CN117706635A - Ghost wave pressing method for low signal-to-noise ratio passive source virtual shot set - Google Patents
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Abstract
The invention belongs to the technical field of marine seismic exploration, in particular relates to a ghost wave compression method aiming at a low signal-to-noise ratio passive source virtual shot set, and solves the problem that the low signal-to-noise ratio in virtual shot set data is different from that of a conventional towing cable observation system. The ghost wave pressing method of the invention achieves the purpose of improving the stability of the method by respectively starting from the two aspects of improving the noise immunity of the ghost wave removing operator and increasing the prior constraint; aiming at the noise interference problem, constructing a data matrix by performing a viewing operation on a data body to meet the condition of introducing sparse constraint, and improving the inversion accuracy by constructing an objective function of a sparse constraint ghost wave square compression method; on the other hand, starting from the stability of the inversion method, the inversion form is changed from the original single-shot circulation to multi-shot simultaneous inversion, and the continuity of the stratum is used as new constraint information in the inversion process, so that the accuracy of the inversion result is further improved, and the applicability of a ghost wave pressure algorithm is greatly expanded.
Description
Technical Field
The invention belongs to the technical field of marine seismic exploration, and particularly relates to a ghost wave pressing method aiming at a passive source virtual shot set with low signal to noise ratio.
Background
Compared with active source exploration, passive source seismic exploration has the advantages of rich low-frequency information, low exploration cost, small environmental damage and the like, can be applied to exploration areas such as cities, environmental protection areas, farmlands and the like which cannot excite active sources, and receives wide attention of the industry. But is limited by the reasons of weak effective signals of passive source data, huge difference between the morphology and active source seismic records and the like, the conventional seismic data processing technology is difficult to apply to the passive source data, and the development of passive source seismic exploration is greatly limited.
In recent years, the development of seismic interferometry has provided new possibilities for passive source exploration development. Seismic interferometry is a completely data-driven method that can construct seismic signals that propagate between any two seismic traces without requiring detailed subsurface a priori information (i.e., convert seismic signals received by two detectors into seismic signals that are excited at one detector and received at the other detector). Therefore, by applying the seismic interference method, the passive source seismic records can be converted into virtual shot sets similar to the active source seismic records. However, in the calculation process of reconstructing the passive source virtual cannon set by using the seismic interferometry, since the passive source distribution condition and the underground structure uniformity cannot meet the basic assumption condition of the seismic interferometry, a large amount of passive source coherent noise is introduced in the virtual cannon set, and the signal to noise ratio of the virtual cannon set is seriously reduced. Although the former research on passive source seismic interference is complete, the noise suppression in the data about the virtual shot set is always unsatisfactory, so that most of the existing mature seismic exploration data processing methods fail, and the ghost wave suppression method in marine seismic data is included.
Ghost waves are a common type of disturbance in marine seismic exploration, the mechanism of which arises because a part of reflected waves are received by detectors by mixing together downstream waves generated via the sea surface with upstream waves generated via the sea bottom, and the two are mutually interfered. The existence of ghost waves can cause problems of reduced on-phase axis resolution, low-frequency loss, notch points in the frequency spectrum and the like of an effective signal, and the subsequent data processing and interpretation processes can be seriously interfered. The ghost wave pressing method in the data domain can be divided into a convolution method and an inversion method, and for the convolution method, the basic assumption is that the delay time of ghost wave removal can be accurately predicted. However, when a large amount of passive source coherent noise exists in the passive source virtual cannon set, the delay time of the ghost wave cannot be accurately calculated, so that the stability of the convolution method is greatly reduced, and the requirement of data processing cannot be met. For inversion algorithms, although accurate ghost wave delay time is not needed, a large amount of coherent noise in the ghost shot set record easily falls into a local extremum to fail, and the inversion accuracy is reduced. Under the condition of low signal-to-noise ratio, the existing ghost wave compression method is easy to cause the problems of in-phase axis distortion, introduction of false frequency and the like in the processing process, and the subsequent data processing and interpretation are seriously affected. At present, in marine passive source seismic exploration, a mature method suitable for low signal-to-noise ratio passive source ghost wave compression is still lacking.
Disclosure of Invention
The invention aims to provide a ghost wave pressing method aiming at a low signal-to-noise ratio passive source virtual gun set, adopts a sparse constraint method, converts an inversion algorithm from original single gun circulation into multi-gun simultaneous inversion, and provides a viewing method aiming at virtual gun set records so as to solve the problems that the low signal-to-noise ratio in virtual gun set data is different from a virtual gun set and a conventional towing cable observation system. The ghost wave pressing effect in the ghost wave pressing method is improved effectively, and the applicability of the ghost wave pressing algorithm is greatly expanded.
The invention aims at realizing the following technical scheme:
a ghost wave pressing method aiming at a low signal-to-noise ratio passive source virtual shot set comprises the following steps:
A. using passive source data T obs Interference reconstruction of serial single track record D v (x i |x j T), reordered to x i Co-shot virtual shot record D as virtual source position s (x i T), and pressShot point position x i The increasing direction is arranged to obtain a 3D data body D (x i ,x j ,t)；
B. For the data volume D (x) i ,x j T) performing a viewing operation according to the common offset direction to change the viewing orientation to obtain data D' (x) off ,x k ,t)；
C. D '(x) is represented in simplified form by D' (t) off ,x k T), carrying out Fourier transformation on the observed data body D '(t) along the time direction to obtain a frequency domain form data body D' (omega) thereof;
D. extracting slices from the frequency domain data body D' (omega) to obtain frequency domain slices, and constructing an objective function F (S) of a sparse constraint ghost square compression method;
E. introducing a priori constraint of formation continuity as a result by introducing low-rank constraint to obtain an objective function containing priori information constraint, thereby obtaining a sparse kernel norm objective function F (S);
F. solving the objective function F (S) obtained in the step E by using a split Bergman iteration strategy, introducing an auxiliary variable matrix Θ, and adjusting the objective function F (S) to F (S, Θ);
G. setting a parameter initial value;
H. iteratively solving, and solving an objective function by utilizing a soft threshold contraction algorithm to obtain an auxiliary matrix Θ;
I. setting Θ=Θ l And solving the objective function by using AMP algorithm to obtain the ghost wave removing result S of the first iteration l ；
J. Judging ghost wave compression result S l Whether convergence conditions are met or not, judging conditions are met, and outputting a ghost wave suppression result S after l iteration processes l The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, resetting parameters, and returning to the step I.
Further, in step A, a series of single track records D v (x i |x j T) is traversing all passive source data T through equation (1) obs Interference acquisition, the cross-correlation interference algorithm in the formula (1) can be used for acquiring the position x of the seismic source i And the detector is located at x j Is the deficiency of (2)Gun set recording channel D v (x i |x j ,t)：
Wherein,representing cross-correlation, D (x i |x j T) and D (x) i |x j T) is the causal and non-causal parts of the single-pass record of the virtual cannon set, the causal part is generally reserved, T is the recording time of the virtual cannon set, T obs Is passive source data, where x i And x j Representing the earth surface receiving position, wherein tau is the receiving time of passive source data, i epsilon (1, n), j epsilon (1, n), and n is the number of receiving channels of the passive source data;
3D data volume D (x i ,x j T) is shown in formula (2):
further, in step B, the transformation process can be expressed as follows:
wherein,
D′ off (x off ,t)＝D(x i ,x j ,t)， (4)
x off ＝x i -x j ， (5)
wherein D' (x) off ,x k T) represents the observed 3D data volume, x off Representing the number of cannons numbered with offset, the range of values [ -x ] n ,x n ]，x k For the detector coordinates, k ε (1, n).
Further, in step C, the data volume D' (ω) is represented by formula (6):
D'(ω)＝FFT[D′(t)] (6)
wherein FFT represents the forward fourier transform operator, ω is the angular frequency.
Still further, in step D, the objective function is formula (7):
wherein S represents data after compression of ghost, lambda is a regularization operator F Representing the Frobenius norm of the solution matrix, S 1 Representing the L1 norm obtained by arranging all elements of the matrix S from the first column.
Further, in step E, the objective function including the prior information constraint is formula (8):
where μ is regularization parameter of low rank constraint, rank represents rank of the matrix, according to matrix completion theory, the following objective function can be used as the optimal convex approximation problem to replace the above formula to obtain formula (9)
Wherein,σ (i) is the ith singular value of the matrix S.
Further, the step F, F (S, Θ) is represented by the formula (10),
where β is an auxiliary parameter in split Bergman iterations, the equation in equation (10) can be solved iteratively by alternately minimizing Θ and S by two sub-problems, according to the split Bergman iteration theory.
Further, in step G, the initial value of the parameter, S 0 ＝S in ，β 0 ＝β in L=1; in step H, the iterative solution process is such that s=s l-1 ，β＝β l-1 And solving an objective function formula (11) by using a soft threshold contraction algorithm:
to obtain the auxiliary matrix Θ, the solving process can be expressed as:
Θ＝T t (S)， (12)
wherein,T t (S) is a soft threshold contraction operator having the following operations,
further, in step I, the objective function is formula (14):
wherein I is an identity matrix,
further, step J is specifically:
judging whether or not the condition is satisfiedWherein ε isBregman iteration stop parameters meet judging conditions and output a suppressed ghost wave result S after one iteration process l The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, let beta l ＝aβ l-1 A e (1, 2), l=l+1, and return to step I.
Compared with the prior art, the invention has the beneficial effects that:
the invention provides a ghost wave pressing method aiming at a ghost wave pressing method with a strong noise interference ghost gun set after interference, which respectively achieves the aim of improving the stability of the method from the two aspects of improving the noise resistance of ghost wave removing operators and increasing priori constraint; aiming at the noise interference problem, constructing a data matrix by performing a viewing operation on a data body to meet the condition of introducing sparse constraint, and improving the inversion accuracy by constructing an objective function of a sparse constraint ghost wave square compression method; on the other hand, starting from the stability of the inversion method, the inversion form is changed from the original single-shot circulation to multi-shot simultaneous inversion, and the continuity of the stratum is used as new constraint information in the inversion process, so that the accuracy of the inversion result is further improved, and the applicability of a ghost wave pressure algorithm is greatly expanded.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings that are needed in the embodiments will be briefly described below, it being understood that the following drawings only illustrate some embodiments of the present invention and therefore should not be considered as limiting the scope, and other related drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a ghost wave compaction method for a low signal-to-noise ratio passive source ghost shot set;
FIG. 2 is a schematic view;
FIG. 3 raw data;
the results of the treatment in fig. 4 are compared, wherein fig. 4a is a conventional method treatment result, and fig. 4b is a modified method treatment result.
Detailed Description
The invention is further illustrated by the following examples:
the invention is described in further detail below with reference to the drawings and examples. It is to be understood that the specific embodiments described herein are merely illustrative of the invention and are not limiting thereof. It should be further noted that, for convenience of description, only some, but not all of the structures related to the present invention are shown in the drawings.
It should be noted that: like reference numerals and letters denote like items in the following figures, and thus once an item is defined in one figure, no further definition or explanation thereof is necessary in the following figures. Meanwhile, in the description of the present invention, the terms "first", "second", and the like are used only to distinguish the description, and are not to be construed as indicating or implying relative importance.
Aiming at the characteristics of passive source virtual cannon set data, the invention starts from the two aspects of improving noise immunity and increasing priori constraint conditions, thereby providing a method suitable for low signal-to-noise ratio passive source virtual cannon set ghost wave compression. Aiming at the problem of low signal to noise ratio in the virtual shot set data, the invention adopts a sparse constraint method to improve the noise resistance of the method; on the other hand, starting from the stability of the inversion method, the method takes the continuity of the stratum as new constraint information in the inversion process, the inversion algorithm is changed from the original single-shot circulation to multi-shot simultaneous inversion, and a viewing method aiming at the virtual shot set record is provided, so that the problem that the virtual shot set is different from a conventional towing cable observation system is solved, and the stability of the inversion algorithm is improved due to the introduction of priori information. The method provided by the invention is respectively from the two angles of the data characteristics of the virtual shot set and the stability of the inversion algorithm, effectively improves the ghost wave pressing effect in the virtual shot set record facing the low signal to noise ratio, and greatly expands the applicability of the ghost wave pressing algorithm.
The invention discloses a ghost wave pressing method aiming at a passive source virtual shot set with low signal to noise ratio, which comprises the following steps:
A. using passive source data T obs Interference reconstruction of serial single track record D v (x i |x j T), reordered to x i Co-firing as virtual source locationPoint virtual cannon record D s (x i T), and according to the shot point position x i The increasing direction is arranged to obtain a 3D data body D (x i ,x j ,t)。
Series single track record D v (x i |x j T) is traversing all passive source data T through equation (1) obs Interference acquisition, the cross-correlation interference algorithm in the formula (1) can be used for acquiring the position x of the seismic source i And the detector is located at x j Virtual cannon set recording track D v (x i |x j ,t)：
Wherein,representing cross-correlation, D (x i |x j T) and D (x) i |x j T) is the causal and non-causal parts of the single-pass record of the virtual cannon set, the causal part is generally reserved, T is the recording time of the virtual cannon set, T obs Is passive source data, where x i And x j Representing the earth surface receiving position, wherein tau is the receiving time of passive source data, i epsilon (1, n), j epsilon (1, n), and n is the number of receiving channels of the passive source data;
3D data volume D (x i ,x j T) is shown in formula (2):
B. for the data volume D (x) i ,x j T) performing a viewing operation according to the common offset direction to change the viewing orientation to obtain data D' (x) off ,x k ,t)。
The process of the transformation can be expressed as follows:
wherein,
D′ off (x off ,t)＝D(x i ,x j ,t)， (4)
x off ＝x i -x j ， (5)
wherein D' (x) off ,x k T) represents the observed 3D data volume, x off Representing the number of cannons numbered with offset, the range of values [ -x ] n ,x n ]，x k For the detector coordinates, k ε (1, n).
C. D '(x) is represented in simplified form by D' (t) off ,x k T), fourier transforming the observed data volume D '(t) in the time direction to obtain the data volume D' (ω) in the form of its frequency domain.
The data volume D' (ω) is represented by equation (6):
D'(ω)＝FFT[D′(t)] (6)
wherein FFT represents the forward fourier transform operator, ω is the angular frequency.
D. And (3) extracting slices from the frequency domain data volume D' (omega) to obtain frequency domain slices, and constructing an objective function F (S) of the sparse constraint ghost square compression method.
The objective function is formula (7):
wherein S represents data after compression of ghost, lambda is a regularization operator F Representing the Frobenius norm of the solution matrix, S 1 Representing the L1 norm obtained by arranging all elements of the matrix S from the first column.
E. And introducing the priori constraint of the formation continuity as a result by introducing the low-rank constraint to obtain an objective function containing the priori information constraint, thereby obtaining the sparse kernel norm objective function F (S).
The objective function containing a priori information constraints is equation (8):
where μ is regularization parameter of low rank constraint, rank represents rank of the matrix, according to matrix completion theory, the following objective function can be used as the optimal convex approximation problem to replace the above formula to obtain formula (9)
Wherein,σ (i) is the ith singular value of the matrix S.
F. Solving the objective function F (S) obtained in the step E by using a split Bergman iteration strategy, introducing an auxiliary variable matrix Θ, and adjusting the objective function F (S) to F (S, Θ).
F (S, Θ) is represented by formula (10),
where β is an auxiliary parameter in split Bergman iterations, the equation in equation (10) can be solved iteratively by alternately minimizing Θ and S by two sub-problems, according to the split Bergman iteration theory.
G. Setting an initial value of a parameter: parameter initial value S 0 ＝S in ，β 0 ＝β in ，l＝1。
H. And (3) carrying out iterative solution, and solving an objective function by utilizing a soft threshold contraction algorithm to obtain an auxiliary matrix theta.
The iterative solving process is that let s=s l-1 ，β＝β l-1 And solving an objective function formula (11) by using a soft threshold contraction algorithm:
to obtain the auxiliary matrix Θ, the solving process can be expressed as:
Θ＝T t (S)， (12)
wherein,T t (S) is a soft threshold contraction operator having the following operations,
I. setting Θ=Θ l And solving the objective function by using AMP algorithm to obtain the ghost wave removing result S of the first iteration l 。
The objective function is equation (14):
wherein I is an identity matrix,
J. judging ghost wave compression result S l Whether convergence conditions are met or not, judging conditions are met, and outputting a ghost wave suppression result S after l iteration processes l The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, resetting parameters, and returning to the step I.
The method comprises the following steps:
judging whether or not the condition is satisfiedWherein epsilon is Bregman iteration stop parameter, meets judgment conditions, and outputs a suppressed ghost wave result S after l times of iteration processes l The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, let beta l ＝aβ l-1 A e (1, 2), l=l+1, and return to step I.
Example 1
The ghost wave compression method aiming at the passive source virtual cannon set with low signal to noise ratio is applied to data test, the data is shown in figure 3, and the sampling time is 2s.
The first step, obtaining the position x of the seismic source by the cross-correlation interference algorithm in the formula (1) i And the detector is located at x j Virtual cannon set recording track D v (x i |x j ,t)：
Wherein,representing cross-correlation, D (x i |x j T) and D (x) i |x j T) is the causal and non-causal parts of the single-pass record of the virtual cannon set, the causal part is generally reserved, and t is the recording time of the virtual cannon set. T (T) obs Is passive source data, where x i And x j Representative of the earth's surface reception location, τ is the time of reception of the passive source data. i epsilon (1, n), j epsilon (1, n), n is the number of receiving channels of passive source data.
Step two, traversing all passive source data T through the formula (1) obs And obtaining a series of single-track records D for interference v (x i |x j T) reordered to x i Co-shot virtual shot record D as virtual source position s (x i T), and according to the shot point position x i The increasing direction is arranged to obtain a 3D data body D (x i ,x j ,t)：
Thirdly, performing a viewing operation (changing the viewing direction) on the data body D obtained in the second step according to the common offset direction to obtain data D', wherein the viewing process can be expressed as follows:
wherein D' off (x off ,t)＝D(x i ,x j ,t)， (4)
x off ＝x i -x j ， (5)
Wherein D' (x) off ,x k T) represents the observed 3D data volume, x off Representing the number of cannons numbered with offset, the range of values [ -x ] n ,x n ]，x k For the detector coordinates, k ε (1, n). The process of the transformation described by equation (3) -equation (5) is shown in fig. 2. The data structure of the virtual shot set data volume D is shown in fig. 2 (a). The trace sets with the same offset in the data volume are connected by different colors, and the result is shown in fig. 2 (b), wherein each connection is the data with the same offset. Rotating D, the viewing angle was shifted from the yellow arrow to the red arrow, and the result was as shown in FIG. 2 (c), at which time the data became 2*n-1 cannon. The data body D in fig. 2 (c) is padded to n tracks along the common offset direction to obtain the observed data D', as in fig. 2 (D), the black part data is 0.
Fourth, D '(t) is used for simplifying and representing D' (x) off ,x k T), fourier transforming the observed data volume D '(t) in the time direction to obtain its frequency domain form D' (ω),
D'(ω)＝FFT[D′(t)]， (6)
wherein FFT represents the forward fourier transform operator, ω is the angular frequency.
Fifthly, in order to improve the anti-noise capability of the ghost wave removing operator G in the passive source virtual gun set with low signal to noise ratio, constructing an objective function of a sparse constraint ghost wave square compression method:
wherein S represents data after compression of ghost, lambda is a regularization operator F Representing the Frobenius norm of the solution matrix, S 1 Representing the L1 norm obtained by arranging all elements of the matrix S from the first column.
Sixth, to improve stability of the inversion process, introducing a priori constraints of formation continuity as a result by introducing low rank constraints, modifying equation (7) to obtain an objective function containing a priori information constraints:
where μ is a regularization parameter of low rank constraint, rank represents the rank of the matrix. According to the matrix completion theory, the following objective function can be used as the optimal convex approximation problem instead of the above formula to obtain formula (9)
Wherein,σ (i) is the ith singular value of the matrix S.
Seventh, by solving the objective function (9) using a split Bergman iterative strategy, introducing the auxiliary variable matrix Θ, equation (9) can be rewritten,
where β is an auxiliary parameter in the split Bergman iteration. According to the split Bergman iteration theory, equation (10) can be solved iteratively by alternately minimizing Θ and S by two sub-problems.
Eighth step, setting S 0 ＝S in ，β 0 ＝β in ，l＝1
Ninth step, set s=s l-1 ，β＝β l-1 And solving an objective function by using a soft threshold contraction algorithm:
to obtain the auxiliary matrix Θ, the solving process can be expressed as:
Θ＝T t (S) (12)
wherein,T t (S) is a soft threshold contraction operator having the following operations,
tenth step, set Θ=Θ l And solving an objective function (14) using an AMP algorithm to obtain a ghost-free result S l
Wherein I is an identity matrix,
eleventh step, judging whether the condition is satisfiedWherein epsilon is Bregman iteration stop parameter, meets judgment conditions, and outputs a suppressed ghost wave result S after l times of iteration processes l . Otherwise, let beta l ＝aβ l-1 A e (1, 2), l=l+1 and back to the ninth step.
As shown in fig. 4, comparing fig. 4a and fig. 4b, it can be seen that the processing results of the embodiments of the present invention have fewer artifacts located at shallow shots, indicating that the method of the present invention provides results with higher stability than the results obtained by the conventional method.
Note that the above is only a preferred embodiment of the present invention and the technical principle applied. It will be understood by those skilled in the art that the present invention is not limited to the particular embodiments described herein, but is capable of various obvious changes, rearrangements and substitutions as will now become apparent to those skilled in the art without departing from the scope of the invention. Therefore, while the invention has been described in connection with the above embodiments, the invention is not limited to the embodiments, but may be embodied in many other equivalent forms without departing from the spirit or scope of the invention, which is set forth in the following claims.
Claims (10)
1. The ghost wave pressing method for the passive source virtual shot set with low signal to noise ratio is characterized by comprising the following steps of:
A. using passive source data T obs Interference reconstruction of serial single track record D v (x i |x j T), reordered to x i Co-shot virtual shot record D as virtual source position s (x i T), and according to the shot point position x i The increasing direction is arranged to obtain a 3D data body D (x i ,x j ,t)；
B. For the data volume D (x) i ,x j T) performing a viewing operation according to the common offset direction to change the viewing orientation to obtain data D' (x) off ,x k ,t)；
C. D '(x) is represented in simplified form by D' (t) off ,x k T), carrying out Fourier transformation on the observed data body D '(t) along the time direction to obtain a frequency domain form data body D' (omega) thereof;
D. extracting slices from the frequency domain data body D' (omega) to obtain frequency domain slices, and constructing an objective function F (S) of a sparse constraint ghost square compression method;
E. introducing a priori constraint of formation continuity as a result by introducing low-rank constraint to obtain an objective function containing priori information constraint, thereby obtaining a sparse kernel norm objective function F (S);
F. solving the objective function F (S) obtained in the step E by using a split Bergman iteration strategy, introducing an auxiliary variable matrix Θ, and adjusting the objective function F (S) to F (S, Θ);
G. setting a parameter initial value;
H. iteratively solving, and solving an objective function by utilizing a soft threshold contraction algorithm to obtain an auxiliary matrix Θ;
I. setting Θ=Θ l And solving the objective function by using AMP algorithm to obtain the ghost wave removing result S of the first iteration l ；
J. Judging ghost wave compression result S l Whether convergence conditions are met or not, judging conditions are met, and outputting a ghost wave suppression result S after l iteration processes l The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, resetting parameters, and returning to the step I.
2. The ghost wave compaction method for a low signal-to-noise ratio passive source ghost shot set according to claim 1, wherein the ghost wave compaction method is characterized by: in step A, series of single track records D v (x i |x j T) is traversing all passive source data T through equation (1) obs Interference acquisition, the cross-correlation interference algorithm in the formula (1) can be used for acquiring the position x of the seismic source i And the detector is located at x j Virtual cannon set recording track D v (x i |x j ,t)：
Wherein,representing cross-correlation, D (x i |x j T) and D (x) i |x j T) is the causal and non-causal parts of the single-pass record of the virtual cannon set, the causal part is generally reserved, T is the recording time of the virtual cannon set, T obs Is passive source data, where x i And x j Representative of the earth's surface receiving location, τ being the passive sourceThe receiving time of the data is i epsilon (1, n), j epsilon (1, n), n is the number of receiving channels of the passive source data;
3D data volume D (x i ,x j T) is shown in formula (2):
3. the ghost wave compaction method for a low signal-to-noise ratio passive source ghost shot set according to claim 2, wherein the ghost wave compaction method is characterized by: in step B, the transformation process can be expressed as follows:
wherein,
D′ off (x off ,t)＝D(x i ,x j ,t)， (4)
x off ＝x i -x j ， (5)
wherein D' (x) off ,x k T) represents the observed 3D data volume, x off Representing the number of cannons numbered with offset, the range of values [ -x ] n ,x n ]，x k For the detector coordinates, k ε (1, n).
4. A ghost wave compaction method for a low signal to noise ratio passive source ghost shot set as claimed in claim 3, wherein: in step C, the data volume D' (ω) is represented by formula (6):
D'(ω)＝FFT[D′(t)] (6)
wherein FFT represents the forward fourier transform operator, ω is the angular frequency.
5. The ghost wave compaction method for a low signal-to-noise ratio passive source ghost shot set as claimed in claim 4, wherein: in step D, the objective function is formula (7):
wherein S represents data after compression of ghost, lambda is a regularization operator F Representing the Frobenius norm of the solution matrix, S 1 Representing the L1 norm obtained by arranging all elements of the matrix S from the first column.
6. The ghost wave compaction method for a low signal-to-noise ratio passive source ghost shot set as claimed in claim 5, wherein: in step E, the objective function containing the prior information constraint is formula (8):
where μ is regularization parameter of low rank constraint, rank represents rank of the matrix, according to matrix completion theory, the following objective function can be used as the optimal convex approximation problem to replace the above formula to obtain formula (9)
Wherein,is the ith singular value of the matrix S.
7. The ghost wave compaction method for a low signal-to-noise ratio passive source ghost shot set as claimed in claim 6, wherein: step F, F (S, Θ) is represented by formula (10),
where β is an auxiliary parameter in split Bergman iterations, the equation in equation (10) can be solved iteratively by alternately minimizing Θ and S by two sub-problems, according to the split Bergman iteration theory.
8. The method for compressing ghost waves of a passive source ghost shot set with a low signal-to-noise ratio according to claim 7, wherein the method comprises the steps of: in step G, the initial value of the parameter S 0 ＝S in ，β 0 ＝β in L=1; in step H, the iterative solution process is such that s=s l-1 ，β＝β l-1 And solving an objective function formula (11) by using a soft threshold contraction algorithm:
to obtain the auxiliary matrix Θ, the solving process can be expressed as:
Θ＝T t (S)， (12)
wherein,T t (S) is a soft threshold contraction operator having the following operations,
9. the method for compressing ghost waves of a passive source ghost shot set with a low signal-to-noise ratio according to claim 8, wherein the method comprises the steps of: in step I, the objective function is formula (14):
wherein I is the unit momentThe array of which is arranged in a row,
10. the method for compressing ghost waves of a passive source ghost shot set with a low signal-to-noise ratio according to claim 9, wherein step J specifically comprises:
judging whether or not the condition is satisfiedWherein epsilon is Bregman iteration stop parameter, meets judgment conditions, and outputs a suppressed ghost wave result S after l times of iteration processes l The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, let beta l ＝aβ l-1 A e (1, 2), l=l+1, and return to step I.
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| CN116203634A (en) * | 2023-03-22 | 2023-06-02 | 吉林大学 | Ghost wave removing method based on low-rank constraint |
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