CN115469364A - Earthquake denoising method and system based on stable Framelet transformation - Google Patents

Earthquake denoising method and system based on stable Framelet transformation Download PDF

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CN115469364A
CN115469364A CN202211299430.9A CN202211299430A CN115469364A CN 115469364 A CN115469364 A CN 115469364A CN 202211299430 A CN202211299430 A CN 202211299430A CN 115469364 A CN115469364 A CN 115469364A
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CN115469364B (en
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吴昊
何良盛
文晓涛
潘晓
王俊钦
韩亚亨
姜财林
马宗楠
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Chengdu Univeristy of Technology
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Abstract

The invention discloses a method and a system for seismic denoising based on stable Framelet transformation, belongs to the field of oil-gas exploration and seismic data processing, and solves the problem of poor seismic denoising effect in the prior art. The method is based on a stable Framelet transformation, a stable Framelet inverse transformation, a haar wavelet construction function and an inverse transformation function; transforming the obtained initial seismic record based on a function, and establishing a seismic denoising target function based on stable Framelet transformation by introducing Lp norm, lagrange multiplier term and dual term; updating the initial seismic record based on an inverse transformation function and an alternating direction multiplier method; and judging whether the values before and after the seismic record updating meet the given conditions, if so, updating the seismic record again by taking the updated seismic record, the Lagrange multiplier term and the dual term as initial values, and otherwise, outputting the updated seismic record. The method is used for seismic denoising.

Description

Earthquake denoising method and system based on stable Framelet transformation
Technical Field
A method and a system for seismic denoising based on stable Framelet transformation are used for seismic denoising, and belong to the field of oil and gas exploration and seismic data processing.
Background
The field seismic record acquisition is influenced by the external environment, so that random noise interference exists in the acquired seismic record, which brings trouble to later explanation work, and particularly the Changtang basin is influenced by noise seriously. Seismic record denoising is therefore a very important process in oil and gas exploration. The seismic record denoising technology based on the transform domain is an important method for seismic denoising, and the principle is to utilize the transform domain to mine the unique characteristics of the seismic record and utilize the characteristics to perform denoising processing so as to achieve the effect of seismic denoising.
The seismic denoising method based on Wavelet transformation is an important method for seismic denoising based on a transformation domain. The method utilizes wavelet transformation to process the seismic record, excavates the sparsity of the seismic record in a wavelet domain, and converts the seismic denoising problem into an optimization problem. And finally, solving by using an optimization theory to obtain a seismic denoising result. Liu Qiong et al, which uses Wavelet transform in seismic denoising of additive noise, demonstrated the feasibility of this approach. However, the Wavelet transform only has one scale function and one Wavelet function, so that the characteristics of the seismic record cannot be fully mined, and meanwhile, redundant information is lost when the Wavelet transform is down-sampled, so that the information can avoid the loss of seismic signals in a mutation area, and the technical problem cannot be solved by other seismic denoising methods. Therefore, a new seismic denoising method needs to be provided to solve the above problems.
In summary, the seismic denoising method based on Wavelet transformation has the following technical problems:
1. the characteristics of the seismic record cannot be fully excavated, so that the seismic denoising effect is poor;
2. redundant information can be lost in sampling, so that the loss of seismic signals in a mutation area is caused, and the seismic denoising effect is poor.
Disclosure of Invention
Aiming at the problems of the research, the invention aims to provide a method and a system for denoising a seismic based on stable Framelet transformation, and solve the problem that the characteristics of seismic records cannot be fully mined in the prior art, so that the seismic denoising effect is poor.
In order to achieve the purpose, the invention adopts the following technical scheme:
a seismic denoising method based on stable Framelet transformation comprises the following steps:
step 1: function W and inverse transform function W for constructing stationary Framelet transform based on stationary Framelet transform, stationary Framelet inverse transform and haar wavelet -1 ；
And 2, step: transforming the obtained initial seismic record based on a function W of stable Framelet transformation, and establishing a seismic denoising target function based on the stable Framelet transformation by introducing Lp norm, a Lagrange multiplier term R and an even term C, wherein the seismic record is a seismic section;
and 3, step 3: based on the inverse transformation function W -1 Updating the initial seismic record by an alternating direction multiplier method to obtain an updated seismic record;
and 4, step 4: judging whether the values before and after the seismic record updating meet the given conditions, if so, taking the updated seismic record, lagrange multiplier term and dual term as initial values, and turning to the step 3, otherwise, outputting the updated seismic record as a final denoising result.
Further, the step 1 comprises the following steps:
step 1.1: construction of a scale function h of a Framelet using a stationary Framelet transform and haar wavelets 1 And 2 wavelet functions h 2 、h 3 The expression is as follows:
Figure RE-GDA0003939919050000021
wherein T represents a transposition of the vector;
step 1.2: using a scale function h 1 And wavelet function h 2 、h 3 The formula of the function W of constructing the stable Framelet transformation, X → G, G is as follows:
G＝[V 1，1 ,V 1，2 ,V 1，3 ,V 2，1 ,V 2，2 ,V 2，3 ,V 3，1 ,V 3，2 ,V 3，3 ] (22)
wherein X represents the seismic record obtained after final denoising, G represents the result of stable Framelet transformation, and V 1，1 ，V 1，2 ， V 1，3 ，V 2，1 ，V 2，2 ，V 2，3 ，V 3，1 ，V 3，2 ，V 3，3 The different components representing the stationary Framelet transform result, component V in row i and column j i，j The formula of (1) is as follows:
Figure RE-GDA0003939919050000022
wherein ,
Figure RE-GDA0003939919050000023
representing a convolution operation;
step 1.3: construction of a scale function for inverse Framelet transformation using stationary inverse Framelet transformation and haar wavelets
Figure RE-GDA0003939919050000024
And 2 wavelet functions
Figure RE-GDA0003939919050000025
The expression is as follows:
Figure RE-GDA0003939919050000026
step 1.4: based on V i，j Scale function of
Figure RE-GDA0003939919050000027
Sum wavelet function
Figure RE-GDA0003939919050000028
Constructing a stationary Framelet inverse transform function W -1 :G→X，W -1 (G) The calculation formula of (a) is as follows:
Figure RE-GDA0003939919050000029
further, the step 2 comprises the following steps:
step 2.1: transforming the input initial seismic record S based on a function W of stable Framelet transformation, performing sparse constraint on the transformed seismic record by adopting Lp (linear regression) norm, and constructing a nonlinear objective function of seismic denoising by combining the seismic record S after the sparse constraint, wherein the objective function is as follows:
Figure RE-GDA0003939919050000031
wherein | | | purple hair F Represents the Frobenius norm,
Figure RE-GDA0003939919050000032
expressing Lp analog norm, wherein p is a constant between 0 and 1, and lambda is a coefficient of a sparse constraint term;
step 2.2: adopting an alternating direction multiplier method, introducing a Lagrange multiplier term R and a dual term C to convert the nonlinear target function into a plurality of linear sub target functions, and obtaining the seismic denoising target function, wherein the expression is as follows:
Figure RE-GDA0003939919050000033
where μ represents the coefficient of the dual constraint term.
Further, the step 3 comprises the following steps:
step 3.1: coefficients λ and μ for a given sparse constraint termAnd initializing seismic record S 0 = S, initial Lagrange multiplier term R 0 =0, initial parity term C 0 ＝0；
Step 3.2: inverse transformation function W based on step 3.1 -1 And seismic denoising objective function J 1 (X) obtaining a new seismic record S using an alternative direction multiplier method and Fermat theorem 1 The formula is as follows:
Figure RE-GDA0003939919050000034
further, the specific steps of step 4 are:
judging whether the values before and after the update of the seismic record meet the given condition | | S 1 -S 0 || 2 /||S 0 || 2 Tol, where tol is a given error term value, and if yes, the updated seismic records, lagrange multiplier term and dual term are taken as initial values, namely S 0 ＝S 1 、 R 0 ＝R 1 、C 0 ＝C 1 And go to step 3, otherwise, output the updated seismic record S 1 And obtaining the final denoising result, namely obtaining the seismic record X obtained after final denoising.
Further, the Lagrange multiplier term R 1 And dual term C 1 The updating steps are as follows:
based on seismic records S 1 And seismic denoising objective function J 2 (R) calculating a new Lagrange multiplier term R by using an alternative direction multiplier method and a soft threshold contraction algorithm 1 The formula is as follows:
R 1 ＝max(|W(S 1 )+C 0 |-(μ/λ) p-2 |W(S 1 )+C 0 | p-1 ,0)·sign(W(S 1 )+C 0 ) (29) where sign represents a sign function, p is a constant between 0-1,. Represents a dot product operation;
based on seismic records S 1 And lagrange multiplier term R 1 Calculating by using an alternative direction multiplier method to obtain a new dual term C 1 Disclosure of the inventionThe formula is as follows:
C 1 ＝C 0 +W(S 1 )-R 1 (30)。
a seismic denoising system based on a stationary Framelet transform, comprising:
the function building module: function W and inverse transform function W for constructing stationary Framelet transform based on stationary Framelet transform, stationary Framelet inverse transform and haar wavelet -1 ；
An objective function construction module: transforming the obtained initial seismic record based on a function W of stable Framelet transformation, and establishing a seismic denoising target function based on the stable Framelet transformation by introducing Lp norm, a Lagrange multiplier term R and an even term C, wherein the seismic record is a seismic section;
an update module: based on the inverse transformation function W -1 Updating the initial seismic record by an alternating direction multiplier method to obtain an updated seismic record;
a denoising judgment module: and judging whether the values before and after the seismic record updating meet the given conditions, if so, taking the updated seismic record, the Lagrange multiplier term and the dual term as initial values, and transferring to an updating module, otherwise, outputting the updated seismic record as a final denoising result.
Further, the specific implementation steps of the function building module are as follows:
step 1.1: construction of a scale function h of a Framelet using a stationary Framelet transform and haar wavelets 1 And 2 wavelet functions h 2 、h 3 The expression is as follows:
Figure RE-GDA0003939919050000041
wherein T represents a transposition of the vector;
step 1.2: using a scale function h 1 And wavelet function h 2 、h 3 The formula of the function W of constructing the stable Framelet transformation, X → G, G is as follows:
G＝[V 1，1 ,V 1，2 ,V 1，3 ,V 2，1 ,V 2，2 ,V 2，3 ,V 3，1 ,V 3，2 ,V 3，3 ] (32)
wherein X represents the seismic record obtained after final denoising, G represents the result of stable Framelet transformation, and V 1，1 ，V 1，2 ， V 1，3 ，V 2，1 ，V 2，2 ，V 2，3 ，V 3，1 ，V 3，2 ，V 3，3 The different components representing the stationary Framelet transform result, component V in row i and column j i，j The formula of (1) is:
Figure RE-GDA0003939919050000042
wherein ,
Figure RE-GDA0003939919050000051
representing a convolution operation;
step 1.3: construction of a scale function for inverse Framelet transformation using stationary inverse Framelet transformation and haar wavelets
Figure RE-GDA0003939919050000052
And 2 wavelet functions
Figure RE-GDA0003939919050000053
The expression is as follows:
Figure RE-GDA0003939919050000054
step 1.4: based on V i，j Scale function of
Figure RE-GDA0003939919050000055
Sum wavelet function
Figure RE-GDA0003939919050000056
Constructing a stationary Framelet inverse transform function W -1 :G→X，W -1 (G) The calculation formula of (a) is as follows:
Figure RE-GDA0003939919050000057
further, the specific implementation steps of the objective function building module are as follows:
step 2.1: transforming the input initial seismic record S based on a function W of stable Framelet transformation, performing sparse constraint on the transformed seismic record by adopting Lp (linear regression) norm, and constructing a nonlinear objective function of seismic denoising by combining the seismic record S after the sparse constraint, wherein the objective function is as follows:
Figure RE-GDA0003939919050000058
wherein | | | purple hair F Represents the Frobenius norm,
Figure RE-GDA0003939919050000059
expressing Lp analog norm, wherein p is a constant between 0 and 1, and lambda is a coefficient of a sparse constraint term;
step 2.2: adopting an alternate direction multiplier method, introducing a Lagrange multiplier term R and a dual term C to convert the nonlinear target function into a plurality of linear sub target functions, and obtaining the seismic denoising target function, wherein the expression is as follows:
Figure RE-GDA00039399190500000510
where μ represents the coefficient of the dual constraint term.
Further, the update module is specifically implemented by the following steps:
step 3.1: given the values of the coefficients λ and μ of the sparse constraint term, and initializing the seismic record S 0 = S, initial Lagrange multiplier term R 0 =0, initial parity term C 0 ＝0；
Step 3.2: inverse transformation function W based on step 3.1 -1 And seismic de-noising targetFunction J 1 (X) obtaining a new seismic record S using an alternative direction multiplier method and Fermat theorem 1 The formula is as follows:
Figure RE-GDA0003939919050000061
the denoising judging module is specifically realized as follows:
judging whether the values before and after the update of the seismic record meet the given condition | | | S 1 -S 0 || 2 /||S 0 || 2 Tol, wherein tol is a given error term value, and if yes, the updated seismic records, lagrange multiplier terms and dual terms are used as initial values, namely S 0 ＝S 1 、 R 0 ＝R 1 、C 0 ＝C 1 And then the earthquake record is converted to an updating module, otherwise, the updated earthquake record S is output 1 Obtaining a final denoising result, namely obtaining a seismic record X obtained after final denoising;
lagrange multiplier term R 1 And dual term C 1 The updating steps are as follows:
based on seismic records S 1 And seismic denoising objective function J 2 (R) calculating a new Lagrange multiplier term R by using an alternative direction multiplier method and a soft threshold contraction algorithm 1 The formula is as follows:
R 1 ＝max(|W(S 1 )+C 0 |-(μ/λ) p-2 |W(S 1 )+C 0 | p-1 ,0)·sign(W(S 1 )+C 0 ) (39) where sign represents a sign function, p is a constant between 0 and 1, represents a dot product operation;
based on seismic records S 1 And lagrange multiplier term R 1 Calculating by using an alternative direction multiplier method to obtain a new dual term C 1 The formula is as follows:
C 1 ＝C 0 +W(S 1 )-R 1 (40)。
compared with the prior art, the invention has the beneficial effects that:
the invention adopts stable Framelet transformation to transform the seismic record, and the introduction of the stable Framelet transformation successfully eliminates the problem of low signal-to-noise ratio caused by the fact that the conventional transform domain denoising method cannot fully excavate the seismic record, thereby achieving the effect of improving the signal-to-noise ratio of the seismic record.
Drawings
FIG. 1 is a block flow diagram of the present invention;
FIG. 2 is a schematic diagram of a seismic section containing noise according to the present invention;
FIG. 3 is a schematic diagram illustrating the denoising result of the seismic section of FIG. 2 according to the present invention;
FIG. 4 is a schematic illustration of the noise removal of FIG. 2 in accordance with the present invention;
FIG. 5 is a diagram illustrating the denoising result of the seismic section of FIG. 2 by using the conventional wavelet transform.
Detailed Description
The invention will be further described with reference to the accompanying drawings and specific embodiments.
The present invention is described in detail below with reference to fig. 1-4.
The technical problems solved by the invention are as follows: the problem that the signal-to-noise ratio is low due to the fact that the conventional transform domain denoising method cannot fully excavate the seismic record is solved, and the effect of improving the signal-to-noise ratio of the seismic record is achieved.
A seismic denoising method based on stable Framelet transformation comprises the following steps:
step 1: function W and inverse transform function W for constructing stable Framelet transform based on stable Framelet transform, stable Framelet inverse transform and haar wavelet -1 (ii) a The method comprises the following specific steps:
step 1.1: construction of a scale function h of a Framelet using a stationary Framelet transform and haar wavelets 1 And 2 wavelet functions h 2 、h 3 The expression is as follows:
Figure RE-GDA0003939919050000071
wherein T represents a transposition of the vector;
step 1.2: using a scale function h 1 Sum wavelet function h 2 、h 3 The formula of the function W of constructing the stable Framelet transformation, X → G, G is as follows:
G＝[V 1，1 ,V 1，2 ,V 1，3 ,V 2，1 ,V 2，2 ,V 2，3 ,V 3，1 ,V 3，2 ,V 3，3 ] (42)
wherein X represents the seismic record obtained after final denoising, G represents the result of stable Framelet transformation, and V 1，1 ，V 1，2 ， V 1，3 ，V 2，1 ，V 2，2 ，V 2，3 ，V 3，1 ，V 3，2 ，V 3，3 The different components representing the stationary Framelet transform result, component V in row i and column j i，j The formula of (1) is:
Figure RE-GDA0003939919050000072
wherein ,
Figure RE-GDA0003939919050000073
representing a convolution operation;
step 1.3: construction of a scale function for inverse Framelet transformation using stationary inverse Framelet transformation and haar wavelets
Figure RE-GDA0003939919050000074
And 2 wavelet functions
Figure RE-GDA0003939919050000075
The expression is as follows:
Figure RE-GDA0003939919050000076
step 1.4: based on V i，j Scale function of
Figure RE-GDA0003939919050000077
He XiaoboFunction(s)
Figure RE-GDA0003939919050000078
Constructing a stationary Framelet inverse transform function W -1 :G→X，W -1 (G) The calculation formula of (a) is as follows:
Figure RE-GDA0003939919050000079
step 2: transforming the obtained initial seismic record based on a function W of stable Framelet transformation, and establishing a seismic denoising target function based on the stable Framelet transformation by introducing Lp norm, a Lagrange multiplier term R and an even term C, wherein the seismic record is a seismic section; the method comprises the following specific steps:
step 2.1: transforming an input initial seismic record S (shown in figure 2) based on a function W of stable Framelet transformation, performing sparse constraint on the transformed seismic record by adopting Lp (linear regression) norm, and constructing a nonlinear objective function of seismic denoising by combining the seismic record S after the sparse constraint, wherein the objective function is as follows:
Figure RE-GDA0003939919050000081
wherein | | | purple hair F Represents the norm of Frobenius,
Figure RE-GDA0003939919050000082
expressing Lp analog norm, wherein p is a constant between 0 and 1, and lambda is a coefficient of a sparse constraint term;
step 2.2: adopting an alternating direction multiplier method, introducing a Lagrange multiplier term R and a dual term C to convert the nonlinear target function into a plurality of linear sub target functions, and obtaining the seismic denoising target function, wherein the expression is as follows:
Figure RE-GDA0003939919050000083
where μ represents the coefficient of the dual constraint term.
And step 3: based on the inverse transformation function W -1 Updating the initial seismic record by an alternating direction multiplier method to obtain an updated seismic record; the method comprises the following specific steps:
step 3.1: given the values of the coefficients λ and μ of the sparse constraint term, and initializing the seismic record S 0 = S, initial Lagrange multiplier term R 0 =0, initial parity term C 0 ＝0；
Step 3.2: inverse transformation function W based on step 3.1 -1 And seismic denoising objective function J 1 (X) obtaining a new seismic record S using an alternative direction multiplier method and Fermat theorem 1 The formula is as follows:
Figure RE-GDA0003939919050000084
and 4, step 4: and (3) judging whether the values before and after the seismic record updating meet the given conditions, if so, taking the updated seismic record, the Lagrange multiplier term and the dual term as initial values, and turning to the step 3, otherwise, outputting the updated seismic record as a final denoising result. The method specifically comprises the following steps:
judging whether the values before and after the update of the seismic record meet the given condition | | S 1 -S 0 || 2 /||S 0 || 2 Tol, wherein tol is a given error term value, and if yes, the updated seismic records, lagrange multiplier terms and dual terms are used as initial values, namely S 0 ＝S 1 、 R 0 ＝R 1 、C 0 ＝C 1 And go to step 3, otherwise, output the updated seismic record S 1 For the final denoising result, the seismic record X obtained after the final denoising is obtained, (as shown in fig. 3), and the schematic diagram of the noise removed in fig. 2 is shown based on the above steps.
Lagrange multiplier term R 1 And dual term C 1 The updating steps are as follows:
based on seismic record S 1 And seismic denoising objective function J 2 (R) multiplication by alternate directionsCalculating by using a submethod and a soft threshold contraction algorithm to obtain a new Lagrange multiplier R 1 The formula is as follows:
R 1 ＝max(|W(S 1 )+C 0 |-(μ/λ) p-2 |W(S 1 )+C 0 | p-1 ,0)·sign(W(S 1 )+C 0 ) (49) where sign represents a sign function, p is a constant between 0-1, represents a dot product operation;
based on seismic records S 1 And lagrange multiplier term R 1 Calculating by using an alternative direction multiplier method to obtain a new dual term C 1 The formula is as follows:
C 1 ＝C 0 +W(S 1 )-R 1 (50)。
in conclusion, the stable Framelet transformation is utilized to process noisy seismic records, lp norm is introduced to construct sparse constraint, and finally, the alternating direction multiplier method is combined to provide the seismic denoising method based on the stable Fremelet transformation. Specifically, as shown in fig. 2 to 5, the black bars represent noise, the abscissa represents the number of seismic traces (Distance/Trace), the ordinate represents the number of Sampling points (Depth/Sampling point), and the denoising result of the seismic profile (two-dimensional seismic data obtained from seismic data) can better remove banded noise compared with the original seismic record, so that the accuracy of the method is proved, and compared with the traditional seismic denoising method based on wavelet transform, the method provided by the invention has a better denoising effect. The invention solves the problems that the noise of the denoising result is still residual and the effective information is lost in the traditional seismic denoising method based on Wavelet transformation, and improves the denoising effect.
The above are merely representative examples of the many specific applications of the present invention, and do not limit the scope of the invention in any way. All the technical solutions formed by the transformation or the equivalent substitution fall within the protection scope of the present invention.

Claims (10)

1. A seismic denoising method based on stable Framelet transformation is characterized by comprising the following steps:
step 1: based on flatFunction W and inverse transform function W for stationary Framelet transform, stationary Framelet inverse transform and haar wavelet construction of stationary Framelet transform -1 ；
Step 2: transforming the obtained initial seismic record based on a function W of stable Framelet transformation, and establishing a seismic denoising target function based on the stable Framelet transformation by introducing Lp norm, a Lagrange multiplier term R and an even term C, wherein the seismic record is a seismic section;
and step 3: based on the inverse transformation function W -1 Updating the initial seismic record by an alternating direction multiplier method to obtain an updated seismic record;
and 4, step 4: and (3) judging whether the values before and after the seismic record updating meet the given conditions, if so, taking the updated seismic record, the Lagrange multiplier term and the dual term as initial values, and turning to the step 3, otherwise, outputting the updated seismic record as a final denoising result.
2. The method for denoising earthquakes based on the stationary Framelet transform as claimed in claim 1, wherein said step 1 comprises the steps of:
step 1.1: construction of a scale function h of a Framelet using a stationary Framelet transform and haar wavelets 1 And 2 wavelet functions h 2 、h 3 The expression is as follows:
Figure RE-FDA0003939919040000011
wherein T represents the transpose of the vector;
step 1.2: using a scale function h 1 And wavelet function h 2 、h 3 The formula of the function W of constructing the stable Framelet transformation, X → G, G is as follows:
G＝[V 1，1 ,V 1，2 ,V 1，3 ,V 2，1 ,V 2，2 ,V 2，3 ,V 3，1 ,V 3，2 ,V 3，3 ] (2)
wherein X represents the mostThe seismic record obtained after final denoising, G represents the result of the smooth Framelet transform, V 1，1 ，V 1，2 ，V 1，3 ，V 2，1 ，V 2，2 ，V 2，3 ，V 3，1 ，V 3，2 ，V 3，3 The different components representing the stationary Framelet transform result, component V in row i and column j i，j The formula of (1) is:
Figure RE-FDA0003939919040000012
wherein ,
Figure RE-FDA0003939919040000013
representing a convolution operation;
step 1.3: scale function for constructing inverse Framelet transforms using stationary inverse Framelet transforms and haar wavelets
Figure RE-FDA0003939919040000014
And 2 wavelet functions
Figure RE-FDA0003939919040000015
The expression is as follows:
Figure RE-FDA0003939919040000021
step 1.4: based on V i，j Scale function of
Figure RE-FDA0003939919040000022
Sum wavelet function
Figure RE-FDA0003939919040000023
Constructing a stationary Framelet inverse transform function W -1 :G→X，W -1 (G) The calculation formula of (a) is as follows:
Figure RE-FDA0003939919040000024
3. the method for seismic denoising based on stationary Framelet transform as claimed in claim 2, wherein said step 2 comprises the steps of:
step 2.1: transforming the input initial seismic record S based on a function W of stable Framelet transformation, performing sparse constraint on the transformed seismic record by adopting Lp (linear regression) norm, and constructing a nonlinear objective function of seismic denoising by combining the seismic record S after the sparse constraint, wherein the objective function is as follows:
Figure RE-FDA0003939919040000025
wherein | | | purple hair F Represents the Frobenius norm,
Figure RE-FDA0003939919040000026
expressing Lp analog norm, wherein p is a constant between 0 and 1, and lambda is a coefficient of a sparse constraint term;
step 2.2: adopting an alternating direction multiplier method, introducing a Lagrange multiplier term R and a dual term C to convert the nonlinear target function into a plurality of linear sub target functions, and obtaining the seismic denoising target function, wherein the expression is as follows:
Figure RE-FDA0003939919040000027
where μ represents the coefficient of the dual constraint term.
4. The method for seismic denoising based on stationary Framelet transform as claimed in claim 3, wherein said step 3 comprises the steps of:
step 3.1: given the values of the coefficients λ and μ of the sparse constraint term, and initializing the seismic record S 0 = S, initial LagThe langri multiplier term R 0 =0, initial parity term C 0 ＝0；
Step 3.2: inverse transformation function W based on step 3.1 -1 And seismic denoising objective function J 1 (X) obtaining a new seismic record S using an alternative direction multiplier method and Fermat theorem 1 The formula is as follows:
Figure RE-FDA0003939919040000028
5. the seismic denoising method based on smooth Framelet transformation as claimed in claim 4, wherein the specific steps of step 4 are:
judging whether the values before and after the update of the seismic record meet the given condition | | S 1 -S 0 || 2 /||S 0 || 2 Tol, wherein tol is a given error term value, and if yes, the updated seismic records, lagrange multiplier terms and dual terms are used as initial values, namely S 0 ＝S 1 、R 0 ＝R 1 、C 0 ＝C 1 And go to step 3, otherwise, output the updated seismic record S 1 And obtaining the final denoising result, namely obtaining the seismic record X obtained after final denoising.
6. The seismic denoising method based on the stationary Framelet transform as claimed in claim 5, wherein: lagrange multiplier term R 1 And dual term C 1 The updating steps are as follows:
based on seismic records S 1 And seismic denoising objective function J 2 (R) calculating a new Lagrange multiplier term R by using an alternative direction multiplier method and a soft threshold contraction algorithm 1 The formula is as follows:
R 1 ＝max(|W(S 1 )+C 0 |-(μ/λ) p-2 |W(S 1 )+C 0 | p-1 ,0)·sign(W(S 1 )+C 0 ) (9)
wherein sign represents a sign function, p is a constant between 0 and 1, and represents a dot product operation;
based on seismic records S 1 And lagrange multiplier term R 1 Calculating by using an alternative direction multiplier method to obtain a new dual term C 1 The formula is as follows:
C 1 ＝C 0 +W(S 1 )-R 1 (10)。
7. a seismic denoising system based on stationary Framelet transform, comprising:
the function building module: function W and inverse transform function W for constructing stationary Framelet transform based on stationary Framelet transform, stationary Framelet inverse transform and haar wavelet -1 ；
An objective function construction module: transforming the obtained initial seismic record based on a function W of stable Framelet transformation, and establishing a seismic denoising target function based on the stable Framelet transformation by introducing Lp norm, a Lagrange multiplier term R and an even term C, wherein the seismic record is a seismic section;
an update module: based on the inverse transformation function W -1 Updating the initial seismic record by an alternating direction multiplier method to obtain an updated seismic record;
a denoising judgment module: and judging whether the values before and after the seismic record updating meet the given conditions, if so, taking the updated seismic record, the Lagrange multiplier term and the dual term as initial values, and transferring to an updating module, otherwise, outputting the updated seismic record as a final denoising result.
8. The seismic denoising system based on the smooth Framelet transform as claimed in claim 7, wherein the function building module is implemented by the following steps:
step 1.1: construction of a scale function h of a Framelet using a stationary Framelet transform and haar wavelets 1 And 2 wavelet functions h 2 、h 3 The expression is as follows:
Figure RE-FDA0003939919040000041
wherein T represents a transposition of the vector;
step 1.2: using a scale function h 1 And wavelet function h 2 、h 3 The formula of the function W of constructing the stable Framelet transformation, X → G, G is as follows:
G＝[V 1，1 ,V 1，2 ,V 1，3 ,V 2，1 ,V 2，2 ,V 2，3 ,V 3，1 ,V 3，2 ,V 3，3 ] (12)
wherein X represents the seismic record obtained after final denoising, G represents the result of stable Framelet transformation, and V 1，1 ，V 1，2 ，V 1，3 ，V 2，1 ，V 2，2 ，V 2，3 ，V 3，1 ，V 3，2 ，V 3，3 The different components representing the stationary Framelet transform result, component V in row i and column j i，j The formula of (1) is:
Figure RE-FDA0003939919040000042
wherein ,
Figure RE-FDA0003939919040000043
representing a convolution operation;
step 1.3: construction of a scale function for inverse Framelet transformation using stationary inverse Framelet transformation and haar wavelets
Figure RE-FDA0003939919040000044
And 2 wavelet functions
Figure RE-FDA0003939919040000045
The expression is as follows:
Figure RE-FDA0003939919040000046
step 1.4: based on V i，j Scale function of
Figure RE-FDA0003939919040000047
Sum wavelet function
Figure RE-FDA0003939919040000048
Constructing a stationary Framelet inverse transform function W -1 :G→X，W -1 (G) The calculation formula of (c) is as follows:
Figure RE-FDA0003939919040000049
9. the method for seismic denoising based on stationary Framelet transform as claimed in claim 8, wherein the objective function constructing module is implemented by the following steps:
step 2.1: transforming the input initial seismic record S based on a function W of stable Framelet transformation, performing sparse constraint on the transformed seismic record by adopting Lp (linear regression) norm, and constructing a nonlinear objective function of seismic denoising by combining the seismic record S after the sparse constraint, wherein the objective function is as follows:
Figure RE-FDA00039399190400000410
wherein | | | calving F Represents the Frobenius norm,
Figure RE-FDA00039399190400000411
expressing Lp analog norm, wherein p is a constant between 0 and 1, and lambda is a coefficient of a sparse constraint term;
step 2.2: adopting an alternate direction multiplier method, introducing a Lagrange multiplier term R and a dual term C to convert the nonlinear target function into a plurality of linear sub target functions, and obtaining the seismic denoising target function, wherein the expression is as follows:
Figure RE-FDA0003939919040000051
where μ represents the coefficient of the dual constraint term.
10. The system for seismic denoising based on stationary Framelet transform of claim 9, wherein the updating module is implemented by the following steps:
step 3.1: given the values of the coefficients λ and μ of the sparse constraint term, and initializing the seismic record S 0 = S, initial Lagrange multiplier term R 0 =0, initial parity term C 0 ＝0；
Step 3.2: inverse transformation function W based on step 3.1 -1 And seismic denoising objective function J 1 (X) obtaining a new seismic record S using an alternative direction multiplier method and Fermat theorem 1 The formula is as follows:
Figure RE-FDA0003939919040000052
the denoising judging module is specifically realized as follows:
judging whether the values before and after the update of the seismic record meet the given condition | | S 1 -S 0 || 2 /||S 0 || 2 Tol, wherein tol is a given error term value, and if yes, the updated seismic records, lagrange multiplier terms and dual terms are used as initial values, namely S 0 ＝S 1 、R 0 ＝R 1 、C 0 ＝C 1 And then the earthquake record is converted to an updating module, otherwise, the updated earthquake record S is output 1 Obtaining a final denoising result, namely obtaining a seismic record X obtained after final denoising;
lagrange multiplier term R 1 And dual term C 1 The updating steps are as follows:
based on seismic records S 1 And seismic denoising objective function J 2 (R), calculating by using an alternative direction multiplier method and a soft threshold shrinking algorithm to obtain a new Lagrange multiplier term R 1 The formula is as follows:
R 1 ＝max(|W(S 1 )+C 0 |-(μ/λ) p-2 |W(S 1 )+C 0 | p-1 ,0)·sign(W(S 1 )+C 0 ) (19)
wherein sign represents a sign function, p is a constant between 0 and 1, and represents a dot product operation;
based on seismic record S 1 And lagrange multiplier term R 1 Calculating by using an alternative direction multiplier method to obtain a new dual term C 1 The formula is as follows:
C 1 ＝C 0 +W(S 1 )-R 1 (20)。
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