CN107728211B - Seismic signal recovery algorithm based on tensor nuclear norm regularization - Google Patents

Seismic signal recovery algorithm based on tensor nuclear norm regularization Download PDF

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CN107728211B
CN107728211B CN201710768733.3A CN201710768733A CN107728211B CN 107728211 B CN107728211 B CN 107728211B CN 201710768733 A CN201710768733 A CN 201710768733A CN 107728211 B CN107728211 B CN 107728211B
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CN107728211A (en
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钱峰
陈�全
张飞笼
胡光岷
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University of Electronic Science and Technology of China
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    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
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Abstract

The invention discloses a seismic signal recovery algorithm based on tensor nuclear norm regularization, which comprises the following steps of: s1, constructing a target function based on Hankel tensor nuclear norm regularization; and S2, solving the objective function through an ADMM algorithm. According to the method, the missing seismic data are reconstructed by a Hankel tensor nuclear norm regularization method according to the characteristics of relevance, redundancy, low rank and the like of pre-stack seismic signals, the algorithm applies the construction mode of a Hankel matrix to tensor nuclear norms by utilizing an HTR-SVD decomposition method, so that the relevance of tensor data blocks is effectively increased, the low rank of the tensor data blocks is improved, the problem that the whole section of the pre-stack seismic signals is missing and cannot be basically restored is solved, the randomly selected work area data can be processed by the method, and the robustness and the adaptability for processing the missing seismic data are improved; the invention can suppress the noise mixed in the signal and improve the signal-to-noise ratio of the signal while filling missing data.

Description

Seismic signal recovery algorithm based on tensor nuclear norm regularization
Technical Field
The invention belongs to the technical field of seismic data signal processing, and particularly relates to a seismic signal recovery algorithm based on tensor nuclear norm regularization.
Background
In the rapid development of world economy, the consumption of resources by various countries of industry and agriculture is also sharply increased, and the demand of non-renewable energy sources such as petroleum and natural gas is continuously increased. Petroleum and natural gas resources are the basis of industrial economy, and whether effective supply of the petroleum and natural gas resources can be guaranteed is related to the aspects of national life, so that the stability and order stability in the country are influenced, and the supply of the petroleum and natural gas resources cannot be interrupted; for this reason, oil storage mechanisms have been established in countries with large energy consumption such as the united states and japan to cope with possible crises.
The oil storage in western regions of China is at high altitude, which brings great difficulty to oil exploration, increases the economic cost correspondingly, and simultaneously has hard imaginable risks in exploration; the relatively deep oil storage position enables the acquisition of exploration data to change to a certain extent, and different rock stratum structures enable seismic data analysts and seismic data interpreters to spend more time, so that different technical methods are required to be applied to reasonable interpretation processing.
In the field earthquake collection process, the placement of the geophone is very difficult due to the blockage of mountains, the obstruction of rivers or the distribution of lakes; the presence of buildings in the vicinity of cities and villages also makes it difficult to place the receiving devices in the corresponding locations; meanwhile, the loss of the transmitting and receiving equipment causes that some places cannot collect underground information, thereby causing the loss of partial data.
Because the capital support is not necessarily sufficient, in consideration of the balance between acquisition and income, a high-density acquisition mode cannot be adopted in the process of acquiring seismic data, only some places are provided with high-density acquisition devices, and some places are provided with a small number of detectors. Thus, part of the acquired seismic data is sparse, and the reflected seismic waves are influenced by each other due to the complex seismic structure, so that multiple waves can appear and various irregular noises are mixed. Due to irregular sampling of the space, an aliasing phenomenon occurs in seismic data recording, and the precision of a wave equation and the effect of filtering clutter are influenced. In the process of evaluating the underground structure, the evaluation has larger deviation due to the loss of partial data, which is not beneficial to the subsequent seismic data interpretation and well drilling and oil pumping work.
The storage of petroleum and natural gas in China is not abundant, the east area with lower altitude and easy petroleum exploration is exploited for a long time, and the east area gradually moves to the west mountainous areas with complex terrain, so that the data acquisition in the east area is difficult, the data quality is poor, and the subsequent processing is difficult. Partial deletion of the pre-stack seismic signals causes disconnection and discontinuity in the formed post-stack data horizon. The noise-mixed prestack data has unclear in-phase axis, unobvious waveform trend, and formed post-stack data is blurred, so that it becomes necessary to reconstruct missing data while suppressing the mixed noise.
Aiming at the condition of missing or mixed noise of the seismic data, the corresponding missing part can be filled through seismic data reconstruction, and various mixed noises can be further effectively suppressed while interpolation is carried out. The reconstruction of the seismic data is a part of the regularization of the seismic data, the reconstruction of the seismic data with noise is effective to suppress the influence of the noise, the reconstruction of the seismic data with partial deletion is to perform interpolation processing on data at a deletion position according to the correlation among the data, and the reconstruction of the seismic data with clutter interference is to remove irrelevant clutter so as to improve the imaging quality.
Seismic data reconstruction makes good use of the fact that sampled seismic data has high redundancy and can be efficiently compressed under appropriate conditions. The rank of the seismic data well describes the redundancy among data, when the waveform similarity of the seismic data is high, a matrix formed by the seismic data is low rank, the rank of a formed tensor is relatively small, and after the data is missing or mixed with noise, the rank of the matrix or the tensor is correspondingly increased.
Although the rank of the tensor well characterizes the correlation information of the seismic data, the rank is not easy to obtain, and a simple and reasonable evaluation operator of a row pair is needed, and is often described by singular values. After singular value decomposition, the singular value with larger numerical value and the corresponding singular vector represent the main information of the matrix; the smaller singular values of the data and the corresponding singular vectors contain less useless information and can be regarded as corresponding parts of noise in decomposition. The method of processing according to singular values is developed more rapidly and proposes a random singular value decomposition to reduce its computation time. According to different dimensionalities of processed seismic data, the method can be divided into two main categories, namely a two-dimensional data matrix filling method and a multi-dimensional data tensor filling method.
The prestack seismic data has similarity between adjacent traces of the same gather, and adjacent gathers have strong correlation, so that the multidimensional seismic data has strong redundancy in space and time, the property can be described by using the low rank property of tensor, and the missing data recovery can be generally attributed to the filling problem of tensor. Methods for solving tensor decomposition generally include CP decomposition, Tucker decomposition, and hovvd decomposition, but these methods do not make good use of the correlation between multidimensional data.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a seismic signal recovery algorithm based on tensor nuclear norm regularization, wherein the algorithm utilizes an HTR-SVD decomposition method to apply a construction mode of a Hankel matrix to tensor nuclear norm, so that the correlation of tensor data blocks is effectively increased, the low rank of the tensor data blocks is improved, and the noise mixed in signals can be suppressed while missing data is filled.
The purpose of the invention is realized by the following technical scheme: the seismic signal recovery algorithm based on tensor nuclear norm regularization comprises the following steps of:
s1, constructing a target function based on Hankel tensor nuclear norm regularization;
and S2, solving the objective function through an ADMM algorithm.
Further, the specific implementation method of step S1 is as follows:
the following objective function was constructed:
Figure GDA0002704546570000031
Figure GDA0002704546570000032
wherein d is an observation signal;
Figure GDA0002704546570000033
to recover the signal;
Figure GDA0002704546570000034
is a sampling operator; when in use
Figure GDA0002704546570000035
If so, the element value is 1, otherwise, the value is 0;
Figure GDA0002704546570000036
nilength representing each dimension, i ═ 1,2 …, N; i | · | purple wind*Representing a nuclear norm; i | · | purple windFRepresents the F (Frobenius) norm;
Figure GDA0002704546570000037
presentation pair
Figure GDA0002704546570000038
Constructing a Hankel tensor nuclear norm;
Figure GDA0002704546570000039
to represent
Figure GDA00027045465700000310
The rank of (d); λ is a loss term (| · | | nonphosphor)F) With a regularization term (| · | | non-conducting phosphor)*) Balance factors (also called regularization parameters) between; r is a self-defined constant;
two convex functions are defined as follows:
Figure GDA00027045465700000311
Figure GDA00027045465700000312
order to
Figure GDA00027045465700000313
The objective function for the augmented lagrange is obtained as:
Figure GDA00027045465700000314
Figure GDA00027045465700000315
representing the inner product of the tensor, d is the observed signal,
Figure GDA00027045465700000316
to recover the signal;
Figure GDA00027045465700000317
is a sampling operator; beta is a penalty operator for balancing the importance of the residual terms.
Further, the specific implementation method of step S2 is as follows: solving the objective function using the ADMM algorithm: solving for fixed other variables
Figure GDA00027045465700000318
Then fixed
Figure GDA00027045465700000319
And
Figure GDA00027045465700000320
will be provided with
Figure GDA00027045465700000321
As a variable; finally updating Lagrange multiplier
Figure GDA00027045465700000322
Solving the objective function J by adopting a method of selecting minimization to obtain:
Figure GDA00027045465700000323
Figure GDA00027045465700000324
Figure GDA00027045465700000325
the three parameters are solved as follows:
A. to is directed at
Figure GDA00027045465700000326
Variables are as follows:
Figure GDA0002704546570000041
order to
Figure GDA0002704546570000042
Obtaining:
Figure GDA0002704546570000043
wherein the content of the first and second substances,
Figure GDA0002704546570000044
in order to map the operator,
Figure GDA0002704546570000045
the following characteristics are also satisfied:
Figure GDA0002704546570000046
order to
Figure GDA0002704546570000047
Obtaining:
Figure GDA0002704546570000048
thus, the objective function J is for a variable
Figure GDA0002704546570000049
The solution to the minimization problem of (1) is:
Figure GDA00027045465700000410
B. to is directed at
Figure GDA00027045465700000411
Variables are as follows:
Figure GDA00027045465700000412
C. to is directed at
Figure GDA00027045465700000413
Variables are as follows:
Figure GDA00027045465700000414
Figure GDA0002704546570000051
the iterative solution of the three parameters specifically comprises the following substeps:
s21, initializing the ranks r of lambda, beta and Hankel matrix, residual threshold tol and maximum iteration number Miter(ii) a Let k equal to 0 and initialize
Figure GDA0002704546570000052
S22, calculation according to the formula (14)
Figure GDA0002704546570000053
S23, calculation according to formula (15)
Figure GDA0002704546570000054
S24, pair
Figure GDA0002704546570000055
Performing singular value decomposition to obtain a singular value set sigma ═ sigma [ sigma ](1),σ(2),...,σ(i),...,σ(M)M represents the total singular value number obtained by singular value decomposition; wherein sigma(i)The value is gradually reduced along with the increase of i and shows a descending trend; if σ(i)Solving by adopting HTR-SVD algorithm when the uniform drop is reduced
Figure GDA0002704546570000056
If σ(i)Solving by adopting an improved HTR-SVD algorithm when the non-uniform decline is detected
Figure GDA0002704546570000057
The HTR-SVD algorithm is specifically realized by the following steps:
s2411, tensor generation
Figure GDA0002704546570000058
Transforming to frequency domain to obtain
Figure GDA0002704546570000059
The frequency domain expression of (a) is:
Figure GDA00027045465700000510
for each of the frequencies j it is possible to,
Figure GDA00027045465700000511
PHa Hankel forward transform operator;
s2412, initializing Gaussian random tensor
Figure GDA00027045465700000512
And transforming it to frequency domain to obtain
Figure GDA00027045465700000513
S2413, changing i to 1;
s2414, according to
Figure GDA00027045465700000514
The following calculations were performed:
Figure GDA00027045465700000515
Figure GDA00027045465700000516
Figure GDA00027045465700000517
wherein, PAAn operational character for Hankel inverse transformation;
s2415, setting i to i +1, and determining whether i is less than n3If yes, returning to the step S2414, otherwise, executing the step S2416;
s2416, according to the obtained
Figure GDA00027045465700000518
To obtain
Figure GDA00027045465700000519
S2417, utilization of
Figure GDA00027045465700000520
To obtain
Figure GDA00027045465700000521
Figure GDA00027045465700000522
The improved HTR-SVD algorithm comprises the following sub-steps:
s2421, calculating
Figure GDA0002704546570000061
In block Hankel form
Figure GDA0002704546570000062
Figure GDA0002704546570000063
The number of columns of (1) is n4；
S2422, constructing Gaussian random tensor
Figure GDA0002704546570000064
p is an oversampling parameter; constructing tensors
Figure GDA0002704546570000065
Figure GDA0002704546570000066
Computing t-QR decompositions
Figure GDA0002704546570000067
S2423, setting i to 1;
s2424, order
Figure GDA0002704546570000068
By using
Figure GDA0002704546570000069
Carrying out t-QR decomposition to obtain
Figure GDA00027045465700000610
S2425, order
Figure GDA00027045465700000611
By using
Figure GDA00027045465700000612
Carrying out t-QR decomposition to obtain
Figure GDA00027045465700000613
S2426, if i is equal to i +1, determining whether i is less than iteration q, if so, returning to step S2424, otherwise, executing S2427;
S2427、
Figure GDA00027045465700000614
wherein
Figure GDA00027045465700000615
Tensor than original
Figure GDA00027045465700000616
Small;
s2428, calculating
Figure GDA00027045465700000617
In block Hankel form
Figure GDA00027045465700000618
Computing
Figure GDA00027045465700000619
Is decomposed and truncated to obtain
Figure GDA00027045465700000620
Figure GDA00027045465700000621
Order to
Figure GDA00027045465700000622
Wherein
Figure GDA00027045465700000623
S2429, calculating
Figure GDA00027045465700000624
Figure GDA00027045465700000625
S25, judgment
Figure GDA00027045465700000626
Or k is less than or equal to MiterIf yes, ending the operation; otherwise, let k be k +1, return to step S22.
Further, the penalty operator β adopts an adaptive updating method, and the algorithm thereof is as follows:
Figure GDA00027045465700000627
wherein, alpha and gamma are constants respectively, wherein gamma is more than 1, and alpha is more than 0 and less than 1.
The invention has the beneficial effects that: according to the method, the missing seismic data are reconstructed by a Hankel tensor nuclear norm regularization method according to the characteristics of correlation, redundancy, low rank and the like of pre-stack seismic signals, and the algorithm applies the construction mode of a Hankel matrix to tensor nuclear norms by using an HTR-SVD decomposition method, so that the correlation of tensor data blocks is effectively increased, and the low rank of the tensor data blocks is improved. Experiments prove that the algorithm can achieve the following beneficial effects:
1. the method provided by the invention can well recover the seismic data with the deletion rate of 50%; with the increase of the deletion rate, although partial deviation exists in the data reconstructed by the method, the data can still be effectively recovered;
2. the method solves the problem that the whole tangent plane of the pre-stack seismic signal is lost and can not be restored basically, so that the method can process randomly selected work area data, and the robustness and the adaptability for processing the lost seismic data are improved;
3. the invention can suppress the noise mixed in the signal while filling the missing data, improve the signal-to-noise ratio of the signal and effectively ensure the structure of the seismic data.
Drawings
FIG. 1 is a flow chart of the algorithm of the present invention;
FIG. 2 is a diagram of a seismic data model in a simulation embodiment of the present invention;
FIG. 3 is a distribution diagram of the rank of seismic data in the frequency domain in a simulation embodiment of the present invention;
FIG. 4 is a cross-sectional view of original data and 70% missing data without noise added in a simulation embodiment of the present invention;
FIG. 5 is a diagram of the reconstruction effect of three algorithms without noise added in the simulation embodiment of the present invention;
FIG. 6 shows the residual error reconstructed by the three algorithms when no noise is added in the simulation embodiment of the present invention;
FIG. 7 is a diagram showing the relationship between the residuals of three algorithms and the number of iterations when no noise is added in the simulation embodiment of the present invention;
FIG. 8 is a graph of the relationship between the residual error and the percentage for the three algorithms without noise added in the simulation example of the present invention;
FIG. 9 is a cut-away view of original data and 70% missing data with noise added in accordance with a simulation embodiment of the present invention;
FIG. 10 is a diagram of the reconstruction effect of three algorithms when noise is added in the simulation embodiment of the present invention;
FIG. 11 shows reconstructed residuals of three algorithms when noise is added in the simulation embodiment of the present invention;
FIG. 12 is a plot of seismic data rank distribution in an actual work area experiment of the present invention;
FIG. 13 is a sectional view of the original data and the data missing 50% in the actual work area experiment according to the present invention;
FIG. 14 is a reconstruction effect diagram of three algorithms in an actual work area experiment of the present invention.
Detailed Description
The method converts the recovery problem of the pre-stack seismic signals into the filling problem of the tensor, and constructs the tensor nuclear norm by using the low rank of the seismic data tensor, so that the missing seismic data can be effectively reconstructed. For solving the tensor filling problem, the invention adopts the ADMM iterative decomposition method, the method extracts an unknown variable and fixes other variables each time, the calculation process is relatively simple, the theoretical basis is strong, and compared with the method of simultaneously updating three parameters at one time, the ADMM algorithm has higher convergence speed and better effect. Meanwhile, aiming at the constructed Hankel tensor nuclear norm, an HTR-SVD decomposition method is provided on the basis of tensor random singular value decomposition (TR-SVD), and a damping truncation selection method of a rank is constructed by combining a matrix rank truncation mode in the damping MSSA. The technical scheme of the invention is further explained by combining specific examples.
As shown in fig. 1, a seismic signal recovery algorithm based on tensor kernel norm regularization includes the following steps:
s1, constructing a target function based on Hankel tensor nuclear norm regularization; the specific implementation method comprises the following steps:
the following objective function was constructed:
Figure GDA0002704546570000071
Figure GDA0002704546570000081
wherein d is an observation signal;
Figure GDA0002704546570000082
to recover the signal;
Figure GDA0002704546570000083
is a sampling operator; when in use
Figure GDA0002704546570000084
If so, the element value is 1, otherwise, the value is 0;
Figure GDA0002704546570000085
nirepresenting the length of each dimension, i 1,2 …, N, e.g. D is a three-dimensional data size of 20 x 30 x 40, then D e R20,30,40；||·||*Representing a nuclear norm; i | · | purple windFRepresents the F (Frobenius) norm;
Figure GDA0002704546570000086
presentation pair
Figure GDA0002704546570000087
Constructing a Hankel tensor nuclear norm;
Figure GDA0002704546570000088
to represent
Figure GDA0002704546570000089
The rank of (d); λ is a loss term (| · | | nonphosphor)F) With a regularization term (| · | | non-conducting phosphor)*) Balance factors (also called regularization parameters) between; r is a self-defined constant; the first line of equation (2) shows
Figure GDA00027045465700000810
Is low-rank;
two convex functions are defined as follows:
Figure GDA00027045465700000811
Figure GDA00027045465700000812
order to
Figure GDA00027045465700000813
The objective function for the augmented lagrange is obtained as:
Figure GDA00027045465700000814
Figure GDA00027045465700000815
representing the inner product of the tensor, d is the observed signal,
Figure GDA00027045465700000816
to recover the signal;
Figure GDA00027045465700000817
is a sampling operator; beta is a penalty operator for balancing the importance of the residual terms.
S2, solving the objective function through an ADMM algorithm; the specific implementation method comprises the following steps: solving for fixed other variables
Figure GDA00027045465700000818
Then fixed
Figure GDA00027045465700000819
And
Figure GDA00027045465700000820
will be provided with
Figure GDA00027045465700000821
As a variable; finally updating Lagrange multiplier
Figure GDA00027045465700000822
Solving the objective function J by adopting a method of selecting minimization to obtain:
Figure GDA00027045465700000823
Figure GDA00027045465700000824
Figure GDA00027045465700000825
the three parameters are solved as follows:
A. to is directed at
Figure GDA00027045465700000826
Variables are as follows:
Figure GDA0002704546570000091
order to
Figure GDA0002704546570000092
Obtaining:
Figure GDA0002704546570000093
wherein，
Figure GDA0002704546570000094
To map an operator, will
Figure GDA0002704546570000095
Therefore, it is not only easy to use
Figure GDA0002704546570000096
The following characteristics are also satisfied:
Figure GDA0002704546570000097
order to
Figure GDA0002704546570000098
Obtaining:
Figure GDA0002704546570000099
thus, the objective function J is for a variable
Figure GDA00027045465700000910
The solution to the minimization problem of (1) is:
Figure GDA00027045465700000911
B. to is directed at
Figure GDA00027045465700000912
Variables are as follows:
Figure GDA00027045465700000913
C. to is directed at
Figure GDA00027045465700000914
Variables are as follows:
Figure GDA00027045465700000915
Figure GDA0002704546570000101
the iterative solution of the three parameters specifically comprises the following substeps:
s21, initializing the ranks r of lambda, beta and Hankel matrix, residual threshold tol and maximum iteration number Miter(ii) a Let k equal to 0 and initialize
Figure GDA0002704546570000102
the tol is initialized by user setting, and when the residual error between the recovery data and the original data is within the threshold tol (namely, the recovery expectation is met), the algorithm terminates the operation;
s22, calculation according to the formula (14)
Figure GDA0002704546570000103
S23, calculation according to formula (15)
Figure GDA0002704546570000104
S24, lemma 1: hypothesis matrix
Figure GDA0002704546570000105
Then the Singular Value Decomposition (SVD) of the matrix a of rank r can be expressed as:
A＝UErV*，Er＝diag({σi}1≤i≤r)
so for the objective function
Figure GDA0002704546570000106
The singular value soft threshold operator is of the form:
S(W)＝U*S(Er)*VT (3-48)
S(Er)＝diag(max((σi-),0)) (3-49)
the singular value approximate solution of the matrix can be derived from theorem 1, but the equation (15) can represent three-dimensional or higher-dimensional seismic data, so that the direct application of the theorem 1 to solve the objective function of the equation (15) has the following problems: (1) the formula (15) introduces the Hankel transformation, so
Figure GDA0002704546570000107
Not merely variables
Figure GDA0002704546570000108
(2) Equation (15) herein deals with high-dimensional seismic data and is limited to two-dimensional seismic data, so there is a problem how to perform SVD decomposition on the high-dimensional data. In order to solve the problems of the two aspects, the invention adopts T-SVD decomposition to carry out singular value decomposition on the high-dimensional data; to solve the constructed Hankel tensor nuclear norm
Figure GDA0002704546570000109
The invention provides an HTR-SVD decomposition algorithm on the basis of TR-SVD.
To pair
Figure GDA00027045465700001010
Performing singular value decomposition to obtain a singular value set sigma ═ sigma [ sigma ](1),σ(2),...,σ(i),...,σ(M)M represents the total singular value number obtained by singular value decomposition; wherein sigma(i)The value is gradually reduced along with the increase of i and shows a descending trend; if σ(i)Solving by adopting HTR-SVD algorithm when the uniform drop is reduced
Figure GDA00027045465700001011
If σ(i)Solving by adopting an improved HTR-SVD algorithm when the non-uniform decline is detected
Figure GDA00027045465700001012
The HTR-SVD algorithm is specifically realized by the following steps:
s2411, tensor generation
Figure GDA00027045465700001013
Transforming to frequency domain to obtain
Figure GDA00027045465700001014
The frequency domain expression of (a) is:
Figure GDA0002704546570000111
for each of the frequencies j it is possible to,
Figure GDA0002704546570000112
PHa Hankel forward transform operator;
s2412, initializing Gaussian random tensor
Figure GDA0002704546570000113
And transforming it to frequency domain to obtain
Figure GDA0002704546570000114
S2413, changing i to 1;
s2414, according to
Figure GDA0002704546570000115
The following calculations were performed:
Figure GDA0002704546570000116
Figure GDA0002704546570000117
Figure GDA0002704546570000118
wherein, PAAn operational character for Hankel inverse transformation;
s2415, setting i to i +1, and determining whether i is less than n3If yes, go back to step S2414, otherwise, go to stepStep S2416;
s2416, according to the obtained
Figure GDA0002704546570000119
To obtain
Figure GDA00027045465700001110
S2417, utilization of
Figure GDA00027045465700001111
To obtain
Figure GDA00027045465700001112
Figure GDA00027045465700001113
The improved HTR-SVD algorithm comprises the following sub-steps:
s2421, calculating
Figure GDA00027045465700001114
In block Hankel form
Figure GDA00027045465700001115
Figure GDA00027045465700001116
The number of columns of (1) is n4；
S2422, constructing Gaussian random tensor
Figure GDA00027045465700001117
p is an oversampling parameter; constructing tensors
Figure GDA00027045465700001118
Figure GDA00027045465700001119
Computing t-QR decompositions
Figure GDA00027045465700001120
S2423, setting i to 1;
s2424, order
Figure GDA00027045465700001121
By using
Figure GDA00027045465700001122
Carrying out t-QR decomposition to obtain
Figure GDA00027045465700001123
S2425, order
Figure GDA00027045465700001124
By using
Figure GDA00027045465700001125
Carrying out t-QR decomposition to obtain
Figure GDA00027045465700001126
S2426, if i is equal to i +1, determining whether i is less than iteration q, if so, returning to step S2424, otherwise, executing S2427;
S2427、
Figure GDA00027045465700001127
wherein
Figure GDA00027045465700001128
Tensor than original
Figure GDA00027045465700001129
Small;
s2428, calculating
Figure GDA00027045465700001130
In block Hankel form
Figure GDA00027045465700001131
Computing
Figure GDA00027045465700001132
Is decomposed and truncated to obtain
Figure GDA00027045465700001133
Figure GDA00027045465700001134
Order to
Figure GDA00027045465700001135
Wherein
Figure GDA00027045465700001136
S2429, calculating
Figure GDA00027045465700001137
Figure GDA0002704546570000121
S25, judgment
Figure GDA0002704546570000122
Or k is less than or equal to MiterIf yes, ending the operation; otherwise, let k be k +1, return to step S22.
Further, the penalty operator β adopts an adaptive updating method, and the algorithm thereof is as follows:
Figure GDA0002704546570000123
wherein α and γ are each independently represented
Wherein the content of the first and second substances,
Figure GDA0002704546570000124
a constraint of the constant alpha is added. Secondary penalty term
Figure GDA0002704546570000125
Is a convex function, which is added to the objective function
Figure GDA0002704546570000126
The convexity of the function is enhanced. Ideally, the residual error
Figure GDA0002704546570000127
It should decrease as the number of iterations increases, however in practice for some reason its value does not always decrease; to achieve this goal, a mandatory change is required
Figure GDA0002704546570000128
Such that it reduces the residual error as the number of iterations increases. The parameters γ and α are therefore introduced such that γ > 1 and 0 < α < 1 are satisfied. With k → ∞, β should tend to be constant.
The invention sets the initialization range of beta to be [2, 10%]This parameter cannot be set initially too large (e.g., 100) because
Figure GDA0002704546570000129
The purpose of (1) is to enhance the convexity of the original objective function. If the parameter β is set too large, the original objective function may not find the optimal value; also the parameter beta cannot be too small, and if it is too small, its effect becomes small, increasing the number of iterations. Based on the results of the experiment, β ═ 2 was set as the optimum choice.
T-SVD line tensor decomposition is introduced aiming at multi-dimensional seismic data, and HTR-SVD decomposition is proposed on the basis, wherein an important parameter, namely the size of a corresponding Hankel matrix rank under each frequency condition, is involved. If the rank r is set to be too large, the seismic data interpolation and denoising effects are not obvious; if the rank r is set too small, energy loss may occur during seismic data recovery. Therefore, the distribution condition of the Hankel matrix rank under the maximum frequency condition is predicted by adopting an experimental method, and the Hankel matrix rank is taken as a parameter of the whole frequency; in order to better estimate the rank of the Hankel matrix, the selected seismic data block cannot be too large.
The validity of the algorithm of the present invention is further verified by experiments below.
1. Simulation experiment
Experimental test environment: the computer is configured as an Inter i3-4170 CPU, 8GRAM, 64-bit Windows 7 operating system; the software used was MATLAB R2014 a.
In the embodiment, firstly, a Rake wavelet is adopted to construct a seismic data model, and three wavelets with frequencies of 30Hz, 40Hz and 50Hz are respectively generated; the wavelets are translated according to different directions to form three types of seismic waves in the sections, and finally the seismic waves of each section are translated to obtain three-dimensional seismic DATA DATA1 with the size of 300X 20, as shown in FIG. 2. In the embodiment, the spatial coordinates of the constructed seismic data are respectively considered as a time direction, an xline direction and an inline direction, wherein the sampling interval of the time direction is 1ms, and the sampling interval is expressed by the number of sampling points, namely 300 sampling points.
To verify the suitability of the constructed seismic data, the rank of the multi-dimensional seismic data needs to be checked. The seismic data are converted into a form of 20 × 300, the third dimension is subjected to DFT conversion, singular values of the seismic data are observed in a frequency domain, and the rank of the seismic data is shown in fig. 3. As can be seen from fig. 3, the singular value has a large mutation at the 3 rd point, and the singular value after the 4 th point is substantially 0, so that the rank of the data may be 3 or 4, and the low rank property is satisfied.
The seismic DATA1 constructed in this embodiment does not have a whole trace missing condition, and therefore, missing processing needs to be performed artificially on the seismic DATA. And randomly selecting missing track numbers in the spaces constructed in the inline direction and the xline direction, and performing missing processing on the selected data whole track in the time direction. The experiment of theoretical data is divided into two parts: the case of no noise and the case of noise addition.
(1) In the absence of noise
When no noise is added to the seismic data, the model data is artificially subjected to deficiency processing, and then the interpolation effect obtained by the algorithm is checked. Selecting and displaying data with total loss of 70%, and displaying 4 sections, namely sections with inline numbers of 1, 5, 10 and 18, wherein the data sections are shown in fig. 4, and fig. 4(a) is an effect graph of the original data sections; FIG. 4(b) is a plot of the effect of a total 70% missing data slice; as can be seen from fig. 4(a), each data slice has a different data format; as can be seen from fig. 4(b), the data in the second section is completely missing, and the other data sections are missing to different extents. Fig. 5 shows the corresponding effect after the missing data is reconstructed, wherein fig. 5(a) shows the reconstruction effect of the algorithm of the present invention, e.g., fig. 5(b) shows the reconstruction effect of the damped MSSA algorithm, and fig. 5(c) shows the reconstruction effect of the T-SVD algorithm. Fig. 6 is a residual error of a corresponding slice after missing data reconstruction, where fig. 6(a) is the residual error of the corresponding slice after algorithm reconstruction of the present invention, fig. 6(b) is the residual error of the corresponding slice after damping MSSA algorithm reconstruction, and fig. 6(c) is the residual error of the corresponding slice after T-SVD algorithm reconstruction. For the effect of the algorithm of the present invention, it can be known from fig. 5(a) that the whole section missing data is completely reconstructed, and the residual error corresponding to fig. 6(a) is very small; fig. 5(b) shows the reconstruction effect of the damped MSSA algorithm, which can also recover the data missing from the whole section, but the data residual reconstructed by this method has larger fluctuation compared with the algorithm of the present invention; fig. 5(c) shows the reconstruction effect of the T-SVD algorithm, the method only uses the kernel norm as the regularization term constraint, and it can be known from the figure that the method cannot reconstruct the situation that the whole tangent plane is missing, and fig. 6(c) shows that the residual ratio of the data with more missing reconstruction of the method is larger.
From quantitative high consideration, the invention respectively compares the convergence conditions of the previous three algorithms. FIG. 7 is a graph of relative residual error versus iteration number for total missing 70% data reconstruction, where the damped MSSA has no iteration processing, so its relative residual error takes the same value; it can be seen from the figure that the convergence speed of the algorithm of the present invention is almost the same as that of the T-SVD, but the relative residual of the algorithm of the present invention can approach smaller values and is smaller than that of the damping MSSA.
When no noise is added to seismic data, the total data are respectively lost by 50%, 70% and 90%, the time complexity of the HTR-SVD algorithm and the time complexity of the Hankel tensor singular value decomposition (HT-SVD) algorithm are compared, the time required by the two algorithms after 50 iterations and the calculation precision (represented by relative residual errors) are respectively calculated, the calculation speed can be increased on the premise of ensuring the calculation precision, and the calculation speed can be increased by about 40%.
Fig. 8 compares reconstruction conditions of different algorithms when the missing data degree is different, and it can be known from the figure that the relative residual error of data reconstruction increases with the increase of the missing rate; the relative residual error of the T-SVD algorithm is larger, the algorithm and the damping MSSA provided by the invention have small difference in relative residual error when the data are missing by less than 30%, but when the data missing degree exceeds 30%, the relative residual error of the algorithm is smaller, and the reconstruction effect is more stable.
(2) Case of adding noise
In order to verify the denoising performance of the algorithm of the present invention, in this embodiment, random noise is added to model data to make the snr of the model data be 3db, 70% of the model data is missing, and a section identical to the section without noise is selected for displaying, where fig. 9(a) and 9(b) respectively show an original data section with the snr of 3db and a data section with the snr of 3db and missing 70%. Fig. 10 and fig. 11 show the denoising reconstruction effect and the residual after reconstruction of the three algorithms, respectively. Fig. 10(a) shows the reconstruction effect of the algorithm of the present invention, fig. 10(b) shows the reconstruction effect of the damping MSSA algorithm, and fig. 10(c) shows the reconstruction effect of the T-SVD algorithm. Fig. 11(a) shows the algorithm-reconstructed residual of the present invention, fig. 11(b) shows the damped MSSA algorithm-reconstructed residual, and fig. 11(c) shows the T-SVD algorithm-reconstructed residual. As can be seen from the figure, the algorithm of the invention not only can well reconstruct a complete signal, but also can improve the signal-to-noise ratio to 11.2 db; the damping MSSA can also reconstruct the outline of the missing data, but a larger residual error is reserved, the signal-to-noise ratio is improved to 6.8db, the suppression noise is not ideal, and the algorithm effect is good without the method; the denoising and reconstruction effects of the T-SVD algorithm are poor.
2. Actual work area test
Selecting DATA of an HBC work area, wherein the DATA collected by the work area is prestack seismic DATA, selecting 601 x 200 x 25 seismic DATA DATA2 from the work area, the spatial directions of the DATA DATA2 are respectively a time direction, an xline direction and an inline direction, 601 is a sampling point in the time direction, and each prestack channel is concentrated with 25 channels of DATA, and the total number of the channels is 200. If the entire DATA is input, it does not satisfy low rank and the required computation time is long, and for this reason DATA2 is divided into smaller DATA blocks, each time a DATA block with size 601 × 25 is selected, and the selected DATA sets are not subjected to overlapping processing.
Fig. 12 shows the distribution of ranks from which the first data block 601 × 25 is selected according to the block division method described above, with 25 data points on the horizontal axis, and the singular value approaches substantially zero after the 10 th data point. To reduce the error of the calculation, the size of the corresponding rank is selected to be 12.
For convenience of display, section data with xline numbers of 1, 5, 8 and 18 are selected, the data basically meet low rank property, fig. 13(a) shows a section of corresponding original data, and it can be known from the figure that the same-phase axes are regular, but the correlation of the data is not as good as that of model data, and the waveform is relatively complex. In FIG. 13(b), in order to determine the case of the corresponding section after 50% of the data is missing, it is known that the data of the second section is missing completely, and the degree of deletion is different in other parts. Fig. 14 shows the reconstruction effect of the three methods, wherein 14(a) is the reconstruction effect of the algorithm of the present invention, fig. 14(b) is the reconstruction effect of the damping MSSA algorithm, and fig. 14(c) is the reconstruction effect of the T-SVD algorithm. As can be seen from the figure, the reconstruction effect of the algorithm can be well close to real data, and the data missing in the whole section can be reconstructed; the damping MSSA cannot well process the condition of the whole section loss, only partial lost signals can be recovered, and the energy is insufficient; the reconstruction effect of the T-SVD is relatively poor; therefore, the algorithm has stronger adaptability and stability.
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. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (3)

1. The seismic signal recovery algorithm based on tensor nuclear norm regularization is characterized by comprising the following steps of:
s1, constructing a target function based on Hankel tensor nuclear norm regularization; the specific implementation method comprises the following steps:
the following objective function was constructed:
Figure FDA0002704546560000011
Figure FDA0002704546560000012
wherein d is an observation signal;
Figure FDA0002704546560000013
to recover the signal;
Figure FDA0002704546560000014
is a sampling operator; when in use
Figure FDA00027045465600000124
If so, the element value is 1, otherwise, the value is 0;
Figure FDA0002704546560000015
nilength representing each dimension, i ═ 1,2 …, N; i | · | purple wind*Representing a nuclear norm; i | · | purple windFRepresents the F norm;
Figure FDA0002704546560000016
presentation pair
Figure FDA0002704546560000017
Constructing a Hankel tensor nuclear norm;
Figure FDA0002704546560000018
to represent
Figure FDA0002704546560000019
The rank of (d); lambda is | · | | non-conducting phosphorF(ii) counting & lt | & gt | & lt | & gt*A balance factor between; r is a self-defined constant;
two convex functions are defined as follows:
Figure FDA00027045465600000110
Figure FDA00027045465600000111
order to
Figure FDA00027045465600000112
The objective function for the augmented lagrange is obtained as:
Figure FDA00027045465600000113
Figure FDA00027045465600000114
representing the inner product of the tensor, d is the observed signal,
Figure FDA00027045465600000115
to recover the signal;
Figure FDA00027045465600000116
is a sampling operator; beta is a penalty operator for balancing the importance of the residual terms;
and S2, solving the objective function through an ADMM algorithm.
2. The seismic signal recovery algorithm based on tensor nuclear norm regularization as set forth in claim 1, wherein the step S2 is specifically appliedThe realization method comprises the following steps: solving the objective function using the ADMM algorithm: solving for fixed other variables
Figure FDA00027045465600000117
Then fixed
Figure FDA00027045465600000118
And
Figure FDA00027045465600000119
will be provided with
Figure FDA00027045465600000120
As a variable; finally updating Lagrange multiplier
Figure FDA00027045465600000121
Solving the objective function J by adopting a method of selecting minimization to obtain:
Figure FDA00027045465600000122
Figure FDA00027045465600000123
Figure FDA0002704546560000021
the three parameters are solved as follows:
A. to is directed at
Figure FDA0002704546560000022
Variables are as follows:
Figure FDA0002704546560000023
order to
Figure FDA0002704546560000024
Obtaining:
Figure FDA0002704546560000025
wherein the content of the first and second substances,
Figure FDA0002704546560000026
to map operators
Figure FDA0002704546560000027
The following characteristics are also satisfied:
Figure FDA0002704546560000028
order to
Figure FDA0002704546560000029
Obtaining:
Figure FDA00027045465600000210
thus, the objective function J is for a variable
Figure FDA00027045465600000211
The solution to the minimization problem of (1) is:
Figure FDA00027045465600000212
B. to is directed at
Figure FDA00027045465600000213
Variables are as follows:
Figure FDA00027045465600000214
Figure FDA00027045465600000215
C. to is directed at
Figure FDA00027045465600000216
Variables are as follows:
Figure FDA0002704546560000031
the iterative solution of the three parameters specifically comprises the following substeps:
s21, initializing the ranks r of lambda, beta and Hankel matrix, residual threshold tol and maximum iteration number Miter(ii) a Let k equal to 0 and initialize
Figure FDA0002704546560000032
S22, calculation according to the formula (14)
Figure FDA0002704546560000033
S23, calculation according to formula (15)
Figure FDA0002704546560000034
S24, pair
Figure FDA0002704546560000035
Performing singular value decomposition to obtain a singular value set sigma ═ sigma [ sigma ](1),σ(2),...,σ(i),...,σ(M)M represents the total singular value number obtained by singular value decomposition; wherein sigma(i)Gradually decreases with the increase of i and gradually decreasesPotential; if σ(i)Solving by adopting HTR-SVD algorithm when the uniform drop is reduced
Figure FDA0002704546560000036
If σ(i)Solving by adopting an improved HTR-SVD algorithm when the non-uniform decline is detected
Figure FDA0002704546560000037
The HTR-SVD algorithm is specifically realized by the following steps:
s2411, tensor generation
Figure FDA0002704546560000038
Transforming to frequency domain to obtain
Figure FDA0002704546560000039
The frequency domain expression of (a) is:
Figure FDA00027045465600000310
for each of the frequencies j it is possible to,
Figure FDA00027045465600000311
PHa Hankel forward transform operator;
s2412, initializing Gaussian random tensor
Figure FDA00027045465600000312
And transforming it to frequency domain to obtain
Figure FDA00027045465600000313
S2413, changing i to 1;
s2414, according to
Figure FDA00027045465600000314
The following calculations were performed:
Figure FDA00027045465600000315
Figure FDA00027045465600000316
Figure FDA00027045465600000317
wherein, PAAn operational character for Hankel inverse transformation;
s2415, setting i to i +1, and determining whether i is less than n3If yes, returning to the step S2414, otherwise, executing the step S2416;
s2416, according to the obtained
Figure FDA0002704546560000041
To obtain
Figure FDA0002704546560000042
S2417, utilization of
Figure FDA0002704546560000043
To obtain
Figure FDA0002704546560000044
Figure FDA0002704546560000045
The improved HTR-SVD algorithm comprises the following sub-steps:
s2421, calculating
Figure FDA0002704546560000046
In block Hankel form
Figure FDA0002704546560000047
Figure FDA0002704546560000048
The number of columns of (1) is n4；
S2422, constructing Gaussian random tensor
Figure FDA0002704546560000049
p is an oversampling parameter; constructing tensors
Figure FDA00027045465600000410
Figure FDA00027045465600000411
Computing t-QR decompositions
Figure FDA00027045465600000412
S2423, setting i to 1;
s2424, order
Figure FDA00027045465600000413
By using
Figure FDA00027045465600000414
Carrying out t-QR decomposition to obtain
Figure FDA00027045465600000415
S2425, order
Figure FDA00027045465600000416
By using
Figure FDA00027045465600000417
Carrying out t-QR decomposition to obtain
Figure FDA00027045465600000418
S2426, if i is equal to i +1, determining whether i is less than iteration q, if so, returning to step S2424, otherwise, executing S2427;
S2427、
Figure FDA00027045465600000419
wherein
Figure FDA00027045465600000420
Tensor than original
Figure FDA00027045465600000421
Small;
s2428, calculating
Figure FDA00027045465600000422
In block Hankel form
Figure FDA00027045465600000423
Computing
Figure FDA00027045465600000424
Is decomposed and truncated to obtain
Figure FDA00027045465600000425
Figure FDA00027045465600000426
Order to
Figure FDA00027045465600000427
Wherein
Figure FDA00027045465600000428
S2429, calculating
Figure FDA00027045465600000429
Figure FDA00027045465600000430
S25, judgment
Figure FDA00027045465600000431
Or k is less than or equal to MiterIf yes, ending the operation; otherwise, let k be k +1, return to step S22.
3. The tensor nuclear norm regularization-based seismic signal recovery algorithm as recited in claim 1 or 2, wherein the penalty operator β adopts an adaptive updating method, and the algorithm is as follows:
Figure FDA00027045465600000432
wherein, alpha and gamma are constants respectively, wherein gamma is more than 1, and alpha is more than 0 and less than 1.
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