CN111158051A - Joint constraint random noise suppression method based on sparse regularization - Google Patents

Joint constraint random noise suppression method based on sparse regularization Download PDF

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CN111158051A
CN111158051A CN202010014167.9A CN202010014167A CN111158051A CN 111158051 A CN111158051 A CN 111158051A CN 202010014167 A CN202010014167 A CN 202010014167A CN 111158051 A CN111158051 A CN 111158051A
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李婧
刘凯
郑彦鹏
刘洋廷
华清峰
李先锋
张林清
赵强
解秋红
马龙
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First Institute of Oceanography MNR
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Abstract

A joint constraint random noise suppression method based on sparse regularization belongs to the technical field of geophysical exploration and specifically comprises the following steps: s1, inputting original seismic records, and constructing a curvelet transform-second-order generalized total variation joint constraint denoising target function according to sparse characteristics of noisy data in a curvelet domain and an image gradient domain; s2, converting the L1-L2 norm regularization model containing the curvelet transformation constraint term into a standard basis pursuit noise reduction problem, inverting the curvelet coefficient with the minimum L1 norm to obtain the seismic record after preliminary noise removal; s3, the preliminary denoising seismic record is used as an input image, the denoising problem of second-order generalized total variation constraint is solved, then the random noise suppression of joint constraint can be achieved, and finally the seismic data with enhanced signal-to-noise ratio is output. The method improves the denoising effect of random noise in the seismic data, and can effectively protect weak signal energy, thereby ensuring the high-quality processing of the subsequent seismic data and the reliability of the seismic geological interpretation result.

Description

Joint constraint random noise suppression method based on sparse regularization
Technical Field
The invention belongs to the technical field of geophysical exploration, and particularly relates to a joint constraint denoising method based on sparse regularization, which is applied to suppression of random noise contained in actual seismic data and protection of weak effective signal energy.
Background
Seismic exploration is one of the conventional means for exploring natural resources such as oil, natural gas and minerals. However, due to the limitations of complex acquisition environments, acquisition technologies and other factors, the seismic data acquired in the field are often accompanied by noise and missing seismic traces, and artifacts caused by inappropriate processing means exist in the seismic data processing process, which affect the final seismic data imaging and even mislead the seismic geological interpretation.
The originally acquired seismic data contains both regular noise and random noise. Existing random noise suppression methods can be broadly classified into 5 categories: denoising methods based on domain transform filtering, such as FX domain filtering, median filtering, beamforming filtering, and the like; denoising methods based on sparse representation, such as wavelet transformation, curvelet transformation, dictionary learning and the like; denoising methods based on matrix rank subtraction, such as singular spectrum analysis, structural low-rank approximation and the like; denoising methods based on signal decomposition, such as empirical mode decomposition, variational mode decomposition, singular value decomposition, and the like; other denoising methods based on novel theories, such as deep learning, mathematical morphology and the like.
In recent years, seismic data reconstruction methods based on transform domain sparse representation have been rapidly developed with the proposal of compressed sensing theory. A prerequisite for using such methods is that the signal itself is sparse or that it can be sparsely represented in a particular transform domain. Therefore, the sparsity description or expression of the signals and the selection of the sparse regularization term are particularly important for the seismic data denoising method. The curvelet transform is a mathematical transform that can provide nearly optimal sparse representation of high dimensional signals, and has good capture capabilities for the geometric features of the seismic wavefield. The denoising method based on curvelet transform can realize random noise suppression and missing seismic data recovery by using curvelet transform according to the difference of three characteristic parameters of random noise and effective signals in scale, angle or position. However, the curvelet transform is a computational harmonic analysis method, and inevitably produces artifact phenomena in the noise suppression result, namely, pseudo gibbs oscillation and a "curvelet-like" pseudo curve are shown, and the interference may introduce a false horizon, which may cause adverse effects on the data denoising quality and main feature identification, and bring difficulty to the feature interpretation of the high-resolution seismic imaging result.
Different from a curvelet transform domain noise suppression method, the widely applied total variation denoising method is an image processing method based on a partial differential equation, a TV-L2 image denoising model is established according to the sparse characteristic of an image gradient domain, the image edge can be kept while noise suppression is carried out, and oscillation is reduced. However, the model has the potential to treat noise as edges, which can result in a blocky effect in image processing, i.e., edge-free sharpening in smooth transition regions in the original image. In order to overcome the blocking phenomenon and avoid edge blurring when an image is subjected to noise reduction, scholars propose a noise reduction method based on second-order generalized total variation, a regularization term of the noise reduction method is equivalent to a mixed regularization term of first-order total variation and second-order total variation, self-adaptive value taking can be realized according to the change of the edge and the detail of the image, and the intensity change of a smooth area of the image can be more accurately described. However, when the second-order generalized total variation regularization constraint random noise suppression method is used alone, if the noise is suppressed excessively, a large amount of important information in an image is lost, otherwise, if the detail information of seismic data is to be retained, it may be difficult to effectively suppress the noise, which results in a poor denoising effect.
Disclosure of Invention
The invention aims to solve the technical problem of providing a joint constraint random noise suppression method based on sparse regularization, which utilizes sparse representation and sparse constraint regularization strategies, adopts curvelet transform and generalized total variation joint constraint conditions, and improves the noise suppression effect on random noise in actual seismic data by constructing and solving a corresponding sparse constraint objective function, so that the method is an effective denoising processing means aiming at the seismic data.
The invention adopts the following technical scheme:
a joint constraint random noise suppression method based on sparse regularization specifically comprises the following steps:
1) setting an artificial seismic source to excite seismic waves, and receiving seismic records x through a surface detector; based on the sparse characteristics of effective signals and noise in different domains in original data x, a target function of joint constrained denoising through curvelet transform-second-order generalized total variation is constructed, and the unconstrained Lagrangian form is as follows:
Figure BDA0002358244440000021
wherein the content of the first and second substances,
Figure BDA0002358244440000031
the value of the variable s when the expression in the brackets reaches the minimum value is expressed; the first term in parentheses is a data fit to ensure that the solution converges continuously toward the true value, where | · |. the luminance is zero2Expressing L2 norm, and being suitable for suppressing Gaussian white noise, wherein x is noisy seismic data, and s is noiseless data to be recovered; the second term and the third term are jointly constrained sparse promoting terms, where | · |. luminous |1Representing the norm L1, C(s) representing the transform coefficients of data s in the curvelet domain,
Figure BDA0002358244440000032
represents the second-order generalized total variation of data s, and mu and η are respectively a curvelet coefficient L1 norm regularization factor and a second-order generalized total variationA total variation regularization factor, wherein a joint denoising result depends on the value of (mu, η);
2) converting a related curvelet transformation constraint denoising part in a joint constraint objective function expressed by the formula (1) into a standard basis tracking denoising problem solution, wherein the expression is as follows:
Figure BDA0002358244440000033
wherein the content of the first and second substances,
Figure BDA0002358244440000034
the value of a variable α when the expression in brackets reaches the minimum value is shown, s.t. shows that an objective function (the former) meets a constraint condition (the latter), epsilon is non-negative arbitrary small quantity, x is noisy seismic data, α is a curvelet transform coefficient set of noiseless data,
Figure BDA0002358244440000035
α is the coefficient of curvelet, N is the total number of curvelet coefficients, CTAn inverse transform operator which is the curvelet transform operator C,
Figure BDA0002358244440000036
for the optimal value of the coefficient of the curvelet,
Figure BDA0002358244440000037
the seismic record after the preliminary denoising is obtained;
selecting a proper parameter epsilon, solving an equation (2) by adopting a spectral projection gradient algorithm, and inverting a group of curvelet coefficients with the minimum L1 norm
Figure BDA0002358244440000038
The coefficient is subjected to curvelet inverse transformation to obtain a seismic record after preliminary denoising
Figure BDA0002358244440000039
The norm of L2 that satisfies the difference with the original data x is less than the selected noise parameter epsilon;
3) recording the seismic data after the preliminary de-noising
Figure BDA00023582444400000310
As an input image, solving a denoising problem of second-order generalized total variation constraint, wherein the expression is as follows:
Figure BDA00023582444400000311
in the discrete case, the equivalent descriptive form of the second order generalized total variation is:
Figure BDA0002358244440000041
wherein the content of the first and second substances,
Figure BDA0002358244440000042
to combine de-noised data, | · | non-woven phosphor2Representing a L2 norm, | · | | non-woven1The norm of L1 is shown,
Figure BDA0002358244440000043
is the gradient of the image s and,
Figure BDA0002358244440000044
Figure BDA0002358244440000045
means that s 'is traversed so that the expression in brackets takes a minimum value, s' means that the value over the image area is a function of the second order symmetric tensor,
Figure BDA0002358244440000046
s′x、s′ythe x-component and y-component of s', respectively,
Figure BDA0002358244440000047
representing the derivative of the variable in the x-direction,
Figure BDA0002358244440000048
representing the derivative of the variable in the y direction, ζ (s ') is the symmetric derivative of s', and weights α and β are positive values.
Selecting a proper regularization factor η, decomposing the formula (3) into subproblems easy to solve through a split Bregman iterative algorithm, and solving through an alternate iterative mode to obtain combined de-noised data
Figure BDA0002358244440000049
Further, in the step (1), in order to construct a joint constraint denoising objective function, specific values of regularization parameters in the formula (1) need to be set with reference to the noise level of seismic data; in practical application, if sufficient effective well control exists, the well position is subjected to quality control to optimize the regular parameters, but if no well data is available, the regular parameter values are usually determined through the experimental results of small-scale data volumes according to the noise level of seismic data.
Further, the step (2) is the interconversion between the standard basis tracking denoising problem and the unconstrained lagrangian form under the condition of containing Gaussian white noise; in the step-by-step solution, the solution (2) of the spectral projection gradient algorithm is preferably selected, so that the value of the parameter epsilon is adjusted only according to the noise-containing estimation, and in other solution modes, a proper non-negative regular parameter mu can be selected for balancing the relationship between the sparsity of the estimation solution and the residual noise of the data.
Further, the step (3) is a second-order generalized total variation denoising problem based on the result obtained in the step (2), and when actually solving, the weight α is set to be 1, only the first derivative and the second derivative in the function are balanced by the weight β, and the default β is 2, which can be applied to most cases.
Compared with the prior art, the invention has the beneficial effects that:
the method disclosed by the invention aims at the sparse characteristics of seismic information in different domains, utilizes the advantages of curvelet transform and second-order generalized total variation, and jointly constrains the optimization problem through the curvelet transform and the second-order generalized total variation, so that the artifact phenomenon which is easy to appear when the curvelet transform is used for denoising singly is avoided, the blocking effect which usually exists when the total variation denoising is adopted singly is avoided, and the denoising effect is obviously better than that of the generalized total variation denoising method. The joint constraint denoising method can effectively suppress noise signals to improve the signal-to-noise ratio of input seismic data to the maximum extent, highlight effective reflection wave in-phase axis energy, well reserve the detail characteristics and the stratum boundary characteristics of underground media, and provide a data basis with high signal-to-noise ratio and high fidelity for subsequent parameter inversion, construction interpretation, reservoir prediction and the like.
Drawings
FIG. 1 is a flow chart of an implementation of a joint constraint random noise suppression method provided by the method of the present invention;
FIG. 2 is a synthetic seismic record without noise (a) and with 40% random noise (signal-to-noise ratio SNR 9.08dB) (b);
fig. 3 is a cross-sectional comparison diagram of model experiments after denoising by five methods, namely, curvelet transform denoising (SNR 10.43dB) (a), total variation denoising (SNR 10.91dB) (b), generalized total variation denoising (SNR 11.05dB) (c), curvelet transform-total variation denoising (SNR 11.87dB) (d), curvelet transform-generalized total variation denoising (SNR 12.59dB) (e);
fig. 4 is a comparison diagram of a single-track record (track 50) of noise-free (a), noise-containing (b), curvelet transform denoising (c), total variation denoising (d), generalized total variation denoising (e), curvelet transform-total variation denoising (f), and curvelet transform-generalized total variation denoising (g) of a model experiment.
Detailed Description
The technical solution of the present invention is further explained by the following embodiments with reference to the attached drawings, but the scope of the present invention is not limited in any way by the embodiments.
According to the sparse characteristics of the seismic data in the curvelet domain and the image gradient domain, sparse representation and sparse regularization strategies are adopted to construct a joint constraint denoising target function, the fidelity term of the joint constraint denoising target function is utilized to ensure that the denoised seismic data can better approximate to the original data, effective detail information is accurately recovered through the joint regularization term, and the edge and discontinuous characteristics in the image are reserved. And (3) preferably selecting proper regularization parameters to solve an objective function, and finally realizing random noise suppression and effective signal weak energy protection so as to improve the signal-to-noise ratio of the seismic data.
Fig. 1 is a flowchart of an implementation of a joint constraint random noise suppression method provided by the method of the present invention, and as shown in fig. 1, the specific implementation process includes:
(1) setting an artificial seismic source to excite seismic waves, and receiving seismic records x through a surface detector; inputting actual seismic records, and constructing a joint constraint denoising target function in an unconstrained Lagrange form by adding a curvelet transformation constraint term and a generalized total variation constraint term according to the sparse characteristics of the actual seismic records in different domains:
Figure BDA0002358244440000061
wherein the content of the first and second substances,
Figure BDA0002358244440000062
the value of the variable s when the expression in the brackets reaches the minimum value is expressed; the first term in parentheses is a data fit to ensure that the solution converges continuously toward the true value, where | · |. the luminance is zero2Expressing L2 norm, and being suitable for suppressing Gaussian white noise, wherein x is noisy seismic data, and s is noiseless data to be recovered; the second term and the third term are jointly constrained sparse promoting terms, where | · |. luminous |1Representing the norm L1, C(s) representing the transform coefficients of data s in the curvelet domain,
Figure BDA0002358244440000063
the second-order generalized total variation of data s is represented, mu and η are respectively a curvelet coefficient L1 norm regularization factor and a second-order generalized total variation regularization factor, and the joint constraint denoising effect is jointly controlled by (mu, η);
(2) aiming at a curvelet transformation constraint denoising part in an objective function, converting the curvelet transformation constraint denoising part into a standard basis pursuit denoising problem (BPDN), wherein the expression is as follows:
Figure BDA0002358244440000064
wherein the content of the first and second substances,
Figure BDA0002358244440000065
when the expression in brackets reaches the minimum valueThe variable α is taken as a value, s.t. shows that an objective function (the former) meets a constraint condition (the latter), epsilon is nonnegative arbitrary small quantity, x is noisy seismic data, α is a curvelet transform coefficient set of noiseless data,
Figure BDA0002358244440000066
α is the coefficient of curvelet, N is the total number of curvelet coefficients, CTAn inverse transform operator which is the curvelet transform operator C,
Figure BDA0002358244440000067
for the optimal value of the coefficient of the curvelet,
Figure BDA0002358244440000068
and recording the seismic data after the initial denoising.
The standard basis tracking noise reduction problem and the unconstrained Lagrange form can be mutually converted under certain conditions, so that the BPDN problem is solved by adopting a spectral projection gradient algorithm and the obtained curvelet coefficient only by adjusting the value of a parameter epsilon according to noise-containing estimation
Figure BDA0002358244440000069
Performing curvelet inverse transformation to obtain a preliminary de-noised seismic record
Figure BDA00023582444400000610
The norm of L2 that satisfies the difference with the original data x is less than the selected noise parameter epsilon;
(3) constructing a second-order generalized total variation (TGV) constraint denoising model based on the result obtained in the step (2):
Figure BDA0002358244440000071
in the discrete case, the equivalent descriptive form of the second order generalized total variation is:
Figure BDA0002358244440000072
wherein the content of the first and second substances,
Figure BDA0002358244440000073
in order to combine the de-noised data,
Figure BDA0002358244440000074
is the gradient of the image s and,
Figure BDA0002358244440000075
Figure BDA0002358244440000076
the derivative of s in the x-direction is represented,
Figure BDA0002358244440000077
the derivative of s in the y-direction is indicated,
Figure BDA0002358244440000078
represents a traversal s 'such that the expression in brackets takes a minimum value, s' represents a function whose value over the image area is a second-order symmetric tensor,
Figure BDA0002358244440000079
s′x、s′yx-component, y-component of s ', respectively, [ zeta ] (s ') is the symmetric derivative of s ',
Figure BDA00023582444400000710
is s'xThe derivative in the x-direction of the signal,
Figure BDA00023582444400000711
is s'xThe derivative in the y-direction of the signal,
Figure BDA00023582444400000712
is s'yThe derivative in the x-direction of the signal,
Figure BDA00023582444400000713
is s'yIn the y-direction, the weights α and β are positive values, the weight α is set to 1, only the first and second derivatives in the function are balanced by the weight β, and the default value β may be 2Is suitable for most situations.
Selecting a proper regularization factor η, decomposing the above formula into subproblems easy to solve through a split Bregman iterative algorithm, and solving through an alternate iterative mode to obtain combined de-noised data
Figure BDA00023582444400000714
The effectiveness of the joint constraint denoising method in suppressing random noise and retaining effective weak signal energy characteristics is verified through the embodiment.
FIG. 2(a) is a seismic record synthesized by convolution of a zero-phase Ricker wavelet with a main frequency of 40Hz and a complex reflection coefficient model, wherein the number of simulated seismic channels is 100, the number of sampling points of each channel is 500, and the sampling interval is 2 ms. The synthetic seismic record of the complex model is close to actual seismic data, the morphological change of the reflection wave event is complex, the energy and the polarity of the reflection wave event are different, and the reflection wave event with weaker energy exists. Fig. 2(b) shows a seismic record with 40% noise, the SNR is 9.08dB, random noise is distributed in the whole seismic section, the energy is stronger, and even the energy of part of the homophase axis of the weak reflected wave is masked.
Five different methods are respectively adopted to process noise-containing data, and it can be known from the effect comparison of fig. 3 and fig. 4 that the combined constraint denoising effect of the curvelet transformation and the generalized total variation is the best, the random noise energy distributed in the seismic section can be effectively suppressed, the signal-to-noise ratio of the denoised seismic section is improved to 12.59dB from 9.08dB, and the effect is obviously better than the other four denoising results (the curvelet transformation denoising SNR is 10.43dB, the total variation denoising SNR is 10.91dB, the generalized total variation denoising SNR is 11.05dB, and the curvelet transformation and the total variation denoising SNR is 11.87 dB). Although the section is cleaner after the curvelet transform denoising, the effective signal is distorted, and an obvious artifact phenomenon is shown in fig. 3 (a); after the total variation denoising, a blocking effect appears in a smooth area of fig. 3 (b); generalized total variation denoising reduces the sharpening effect of non-edge positions, but the weak energy in-phase axis is suppressed, as shown in fig. 3(c)1200-1300 ms. Compared with the combined denoising method of curvelet transformation and total variation, the method not only retains weaker reflected wave energy to a certain extent, so that the method can still retain better similarity with the original signal after denoising, such as weak reflection signals near 700ms and 750ms shown in fig. 4, but also has better suppression effect on random noise of non-effective signal segments than the former.
The invention provides a joint constraint random noise suppression method based on sparse regularization, which comprises the following steps: effectively representing anisotropic characteristics such as edges, curves and the like by utilizing curvelet transformation; different image areas are selectively processed by utilizing second-order generalized total variation, so that edge detail information can be reserved, and blocking effect is avoided to a great extent. The numerical simulation denoising result shows that the method can well protect the weak reflection energy characteristics of effective signals while effectively suppressing random noise, highlight the horizon relation of various underground stratums and be beneficial to explaining the geological structure of a research and exploration area.

Claims (4)

1. A joint constraint random noise suppression method based on sparse regularization is characterized by specifically comprising the following steps:
1) setting an artificial seismic source to excite seismic waves, and receiving seismic records x through a surface detector; based on the sparse characteristics of effective signals and noise in different domains in original data x, a target function of joint constrained denoising through curvelet transform-second-order generalized total variation is constructed, and the unconstrained Lagrangian form is as follows:
Figure FDA0002358244430000011
wherein the content of the first and second substances,
Figure FDA0002358244430000012
the value of the variable s when the expression in the brackets reaches the minimum value is expressed; the first term in parentheses is a data fit to ensure that the solution converges continuously toward the true value, where | · |. the luminance is zero2Expressing L2 norm, and being suitable for suppressing Gaussian white noise, wherein x is noisy seismic data, and s is noiseless data to be recovered; the second term and the third term are jointly constrained leanA thinning promotion term in which | · |. non-woven phosphor1Representing the norm L1, C(s) representing the transform coefficients of data s in the curvelet domain,
Figure FDA0002358244430000013
the second-order generalized total variation of the data s is represented, mu and η are respectively a curvelet coefficient L1 norm regularization factor and a second-order generalized total variation regularization factor, and the joint denoising result depends on the value of (mu, η);
2) converting a related curvelet transformation constraint denoising part in a joint constraint objective function expressed by the formula (1) into a standard basis tracking denoising problem solution, wherein the expression is as follows:
Figure FDA0002358244430000014
wherein the content of the first and second substances,
Figure FDA0002358244430000015
the value of a variable α when the expression in brackets reaches the minimum value is shown, s.t. shows that an objective function (the former) meets a constraint condition (the latter), epsilon is non-negative arbitrary small quantity, x is noisy seismic data, α is a curvelet transform coefficient set of noiseless data,
Figure FDA0002358244430000016
α is the coefficient of curvelet, N is the total number of curvelet coefficients, CTAn inverse transform operator which is the curvelet transform operator C,
Figure FDA0002358244430000017
for the optimal value of the coefficient of the curvelet,
Figure FDA0002358244430000018
the seismic record after the preliminary denoising is obtained;
selecting a proper parameter epsilon, solving an equation (2) by adopting a spectral projection gradient algorithm, and inverting a group of curvelet coefficients with the minimum L1 norm
Figure FDA0002358244430000019
The coefficient is subjected to curvelet inverse transformation to obtain a seismic record after preliminary denoising
Figure FDA00023582444300000110
The norm of L2 that satisfies the difference with the original data x is less than the selected noise parameter epsilon;
3) recording the seismic data after the preliminary de-noising
Figure FDA0002358244430000021
As an input image, solving a denoising problem of second-order generalized total variation constraint, wherein the expression is as follows:
Figure FDA0002358244430000022
in the discrete case, the equivalent descriptive form of the second order generalized total variation is:
Figure FDA0002358244430000023
wherein the content of the first and second substances,
Figure FDA0002358244430000024
to combine de-noised data, | · | non-woven phosphor2Representing a L2 norm, | · | | non-woven1The norm of L1 is shown,
Figure FDA0002358244430000025
is the gradient of the image s and,
Figure FDA0002358244430000026
Figure FDA0002358244430000027
means that s 'is traversed so that the expression in brackets takes a minimum value, s' means that the value over the image area is a function of the second order symmetric tensor,
Figure FDA0002358244430000028
s′x、s′ythe x-component and y-component of s', respectively,
Figure FDA0002358244430000029
representing the derivative of the variable in the x-direction,
Figure FDA00023582444300000210
represents the derivative of the variable in the y direction, ζ (s ') is the symmetric derivative of s', and weights α and β are positive values;
selecting a proper regularization factor η, decomposing the formula (3) into subproblems easy to solve through a split Bregman iterative algorithm, and solving through an alternate iterative mode to obtain combined de-noised data
Figure FDA00023582444300000211
2. The method as claimed in claim 1, wherein in the step (1) of constructing the joint constraint denoising objective function, the concrete values of the regularization parameters in the formula (1) need to be set by referring to the noise level of the seismic data; in practical application, if sufficient effective well control exists, quality control is carried out on well positions to optimize the regular parameters, but if no available well data exists, the regular parameter values need to be determined through the experimental results of the small-scale data volume according to the noise level of seismic data.
3. The method according to claim 1, wherein the step (2) is a mutual conversion of the standard basis pursuit denoising problem and the unconstrained lagrangian form under the condition of containing gaussian white noise; the solution (2) is solved by using a spectral projection gradient algorithm, and the value of the parameter epsilon is adjusted only according to the noisy estimation.
4. The method according to claim 1, wherein the step (3) is a second-order generalized total variation denoising problem based on the result obtained in the step (2), and when actually solving, the weight α is set to 1, and only the first derivative and the second derivative in the function are balanced by the weight β, and the default β is set to 2.
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