CN117434606B - Seismic section denoising method based on improved Laplace filtering reverse time migration - Google Patents
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Abstract
The seismic section denoising method based on improved Laplace filtering reverse time migration comprises the steps of collecting seismic data, obtaining RTM imaging results, and denoising by using the improved Laplace filtering. Aiming at the problems that the application of the Laplace filtering always has in-phase axis discontinuity and noise residue on an imaging section, the invention provides a Laplace filtering operator which only selects the second-order differential approximate normal direction component and omits the second-order differential approximate parallel direction component to carry out filtering processing on the basis of analyzing a low-frequency noise generation mechanism, the basic principle of a Laplace filtering method and the problems, and verifies the theoretical feasibility of the Laplace filtering operator in a wave number domain. The simple and complex model data trial calculations show that the improved Laplacian operator has obvious filtering effect, and can better eliminate imaging noise generated by back reflection waves and simultaneously maintain the in-phase axis continuity and amplitude characteristic of effective signals.
Description
Technical Field
The invention relates to a denoising method of an earthquake section based on improved Laplace filtering reverse time migration, which is used for solving the problems of non-continuity of a phase axis and noise residue on an imaging section after Laplace filtering is applied after migration, and belongs to the field of exploration geophysics.
Background
The prestack depth migration algorithm can be divided into two methods, ray-based and wave equation-based. Single pass wave equation migration does not accurately image steep dip interfaces, and kirchhoff integration (Zhu and Lines, 1998) can only describe the wave propagation in smooth media, making it difficult to obtain accurate imaging profiles of complex construction. Reverse time migration is a seismic data processing method that extends migration of seismic reflection data based on a two-way wave equation to obtain subsurface images that effectively describe geologic structures (Baysal et al, 1983; loewenthal et al, 1983). The method has the advantages that various waveforms such as reflected waves, multiple waves, revolving waves and the like can be utilized, and the method is not limited by stratum dip angles and the like. In the study of imaging conditions of reverse time migration, the cross-correlation imaging conditions are simple to implement and are most widely applied, which can provide accurate dynamic characteristics for imaging (Claerbout, 1971), but compared with the single-pass wave migration method, the application of the cross-correlation imaging process can lead to the existence of a large amount of strong amplitude low-frequency noise generated by back-reflected waves (Liu et al, 2010; du et al, 2013), whether acoustic waves, elastic waves (Yan and Sava, 2008; du et al, 2012) and anisotropic reverse time migration (Zhang et al, 2009), which seriously affect the quality of imaging profile and further processing and interpretation. The back reflected wave refers to the reflected wave generated when the wave encounters a strong wave impedance interface in the wave field of the seismic source or the wave field of the detector in the extending process, and the cross-correlation with the wave which normally propagates can generate strong amplitude low-frequency noise on the imaging section.
Since the 80 s of the 20 th century, various methods have been proposed by many scholars at home and abroad for suppressing reverse time offset low frequency noise. In 1983, baysal proposed that in post-stack RTM, a two-way reflectionless wave equation is obtained by matching the reflection at the wave impedance suppression interface to suppress low frequency noise; while Loewenthal et al (1987) is a method of reducing back reflection by smoothing the velocity model, both methods do not work well in the pre-stack case. In recent years, an imaging method based on traveling wave separation is proposed (Yoon and Marfurt,2006; liu et al, 2011), all wave field types are fully utilized, an upstream wave and a downstream wave are obtained by separation according to a geometric relation between an incident wave and a reflected wave, cross-correlation is carried out between components in different vertical directions in the wave field to obtain an imaging result, and low-frequency noise generated by back reflection is effectively suppressed. In 2015 Fei et al (2015) proposed to preserve only the up-going wave of the source wavefield and the down-going wave of the detector wavefield, both applied to cross-correlation imaging conditions to suppress low frequency noise and migration artifacts. Wang et al (2016) further proposed separating the wavefield into left, right, up and down traveling wavefields, and cross-correlating the components of the source and detector wavefields, respectively, to effectively separate low frequency noise and imaging crosstalk. The least squares offset (nemet al, 1999; dai et al, 2011; hu et al, 2016) is one direction of development of conventional offset, the core idea of which is to solve for an exact solution of the model space under linear inversion theory, equivalent to taking into account amplitude compensation and waveform correction on the basis of conventional offset construction imaging, resulting in more amplitude-preserving and higher resolution imaging results. Later, combining the least squares idea and reverse time shift method results in a least squares reverse time shift (Dong et al 2012; zhang et al 2015; liu et al 2016,2017) that provides reflected wave imaging results with more balanced amplitude and higher resolution and eliminates offset noise than conventional shift methods. However, least squares reverse time shifting typically requires multiple iterations to obtain a converging final image, placing significant demands on computational cost and speed. Correspondingly, the techniques such as the Poynting vector and the like are also introduced into the least square reverse time offset (Yang and Zhang,2018; wang et al, 2021), so that the efficiency of the least square algorithm is improved, and good effects are obtained.
Disclosure of Invention
The invention aims to provide an improved Laplace filtering reverse time migration method, which aims to solve the problems of non-continuity of a same phase axis and noise residual existing in the use of Laplace filtering after migration.
The Laplace operator can better eliminate imaging noise, but also changes the amplitude and phase characteristics of the same phase axis of the seismic profile, thus requiring additional correction. Laplace can be modified to suppress low frequency noise in order to obtain higher quality imaging profiles. Aiming at the fact that the Laplace operator cannot better protect the phase axis and the phase characteristic of the seismic section, the invention provides an improved Laplace filtering algorithm, and the high-quality seismic section can be obtained while imaging noise is suppressed.
The seismic section denoising method based on improved Laplace filtering reverse time migration is characterized by comprising the following steps of:
1) Collecting seismic data by using a detector;
2) Obtaining a source wave field and a detector wave field by using a reverse time migration method, and obtaining an imaging result after RTM processing by applying a zero-delay cross-correlation imaging condition:
the zero-delay cross-correlation imaging condition is expressed as
(one)
Wherein%x,z) The position of the imaging point is indicated,T max representing the maximum recording time, S and R are the source wavefield and the detector wavefield,I(x,z) Representing imaging results after RTM processing;
3) And (3) filtering:
suppressing low-frequency noise after imaging the imaging result by using a Laplace filter, wherein the Laplace filter is expressed as
(II)
In the middle ofI lap Representing imaging results after Laplace filteringx,z) The position of the imaging point is indicated,I(x,z) Representing the imaging result after RTM processing, laplacian 2 Is a second order differential operator in the n-dimensional Euclidean space, laplace operator is defined as
(III)
Carrying the formula (I) into the formula (II) to obtain an imaging result after Laplace filtering, wherein the imaging result is expressed as
(IV)
The derivative of the discrete function is calculated in a differential mode, and the Laplace operator is respectively calculatedx、zLaplace operator filter for obtaining discrete function by differentiating second derivatives of two directions, which is expressed as
(V)
The eight-neighborhood second-order differential Laplace filter is split, and as the two opposite directions in the eight neighborhood have the same mathematical meaning, the eight-neighborhood second-order differential Laplace filter is split into four Laplace filters to obtain improved Laplace filtering:(six)
(seven)
(eight)
(nine)
And (3) operating all imaging results after RTM by using any one modified Laplace filter shown in formulas (six), (seven), (eight) and (nine), so as to denoise the seismic section.
In the step 2), the cross-correlation imaging condition is modified by using the normalization of the source wave field, and the obtained imaging result is expressed as:
(ten)
In the step 2), the RTM process includes a conventional acoustic wave RTM or an elastic wave RTM.
In the step 3), due toI 1 lap AndI 3 lap is symmetrical in mathematical sense, has smaller meaning for denoising the seismic section, and is preferably selected fromI 2 lap AndI 4 lap the improved laplacian filter is shown.
In the step 3), the method is preferably selected fromI 2 lap The improved laplacian filter is shown.
An improved method of laplace filtering is presented and implemented herein based on the application of laplace filtering in reverse time offset imaging. The reverse time migration is a migration imaging method based on double-pass waves, has the advantages of high imaging precision of complex structures, and the like, and is widely applied. The zero-delay cross-correlation imaging condition is a simple and effective imaging condition in reverse time migration, but the cross-correlation between the back reflection wave and the wave of the wave field of detection can generate a large amount of low-frequency noise with strong amplitude.
There is therefore further proposed an improved method of laplace filtering, unlike laplace filtering, in which, in the approximation of the second derivative, it is further deduced that only the component related to the second derivative in the z-axis direction is considered, the component related to the second derivative in the x-axis direction is ignored, the component related to the second derivative in the x-axis direction is more sensitive to the tilt configuration, and the component related to the second derivative in the z-axis direction has a better response to the parallel structure or the tilt configuration, and offset artefacts can be further removed. And theoretical analysis is performed in the wave number domain, which proves that the method is feasible.
The Marmousi model data trial calculation result shows that the effectiveness of Laplace filtering is improved, the continuity of the phase axis is ensured, and meanwhile, low-frequency noise generated by back reflection is suppressed, so that an imaging result with good amplitude balance and higher signal-to-noise ratio is obtained. The method for applying improved Laplace filtering after reverse time migration is simple and direct, is used for filtering the offset data body, is only related to the grid size of the data body, and has the advantages of high calculation efficiency and strong practicability.
Drawings
FIG. 1 is a schematic diagram of a source wavefield vector and a detector wavefield vector;
FIG. 2 is a reverse time offset low frequency noise generation scheme;
FIG. 3 is a reverse time migration result under full-wave equation conditions;
FIG. 4 is a modified Laplace filterI 1 lap Results of (2);
FIG. 5 is a modified Laplace filterI 2 lap Results of (2);
FIG. 6 is a modified Laplace filterI 3 lap Results of (2);
FIG. 7 is a modified Laplace filterI 4 lap Results of (2);
FIG. 8 is a flow chart of denoising a seismic section based on improved Laplace filtering reverse time migration as shown in example 1;
FIG. 9 is a schematic diagram of a Sigsbee2 model;
FIG. 10 is a plot of post-denoising imaging result of a seismic section based on improved Laplace filtering reverse time migration as shown in example 1.
Detailed Description
The seismic section denoising method based on improved Laplace filtering reverse time migration comprises the following steps: 1) Seismic data is acquired using detectors.
2) The source wavefield and the detector wavefield are obtained using a reverse time migration method, and a conventional reverse time migration algorithm applies zero-delay cross-correlation imaging conditions to obtain imaging (i.e., imaging results after RTM):
the zero-delay cross-correlation imaging condition can be expressed as
（1）
Wherein, the method comprises the following steps ofx,z) The position of the imaging point is indicated,T max the maximum recording time is indicated and the time of the recording is indicated,SandRis the source wavefield and the detector wavefield,Irepresenting imaging results after RTM. Such imaging conditions are simple to apply and provide a stable structural image. However, the image amplitude of the cross-correlation imaging has no explicit physical relationship with the reflection coefficient, but scales arbitrarily depending on the source intensity and receiver coverage. Normalization of the source wavefield may be used to modify the cross-correlation imaging conditions, and the resulting imaging results may be expressed as:
（2）
during reverse time migration, low frequency noise of strong amplitude tends to contaminate the migration results, especially at the upper portion of the shallow and strong reflection interfaces. When back-reflection develops, noise is generated during the cross-correlation of the source and detector wavefields due to the cross-correlation between the forward propagating source and back-reflected detector wavefields and the back-reflected source and forward propagating detector wavefields, which noise exhibits low frequency characteristics in the frequency domain. The lower frequency noise produces strong interference in the offset profile, and the greater the difference in wave impedance between the interfaces, the more energy the back reflected wave energy, and the more low frequency noise. Compared with the single-pass wave migration method, the single-pass wave migration method limits the propagation direction of the wave field, the wave field which is forward extended by the shot points encounters the reflecting surface to only consider downward propagation, so that imaging only occurs in the direction opposite to the propagation direction of the shot points, and therefore the single-pass wave has no strong energy noise.
3) And (3) filtering:
after conventional acoustic wave RTM or elastic wave RTM is carried out on the imaging result, the Laplace filter is used for suppressing low-frequency noise after imaging, and the Laplace filter is expressed as
（3）
In the middle ofI lap Representing imaging results after Laplace filteringx,z) The position of the imaging point is indicated,I(x,z) Representing the imaging result after RTM processing (namely, the seismic section after RTM), laplacian 2 Is a second order differential operator in the n-dimensional Euclidean space, laplace operator is defined as
（4）
Taking formula (2) into formula (3) to obtain Laplace filtered imaging result, expressed as
（5）
The derivative of the discrete function can be calculated in a differential mode, and the derivative can be respectively calculated for Laplace operatorsx、zLaplace operator filter for obtaining discrete function by differentiating second derivatives of two directions, which is expressed as
（6）
The improved Laplace filtering is characterized in that an eight-neighborhood second-order differential Laplace filter is split, and the two opposite directions in the eight neighborhood have the same mathematical meaning, so that the eight-neighborhood second-order differential Laplace filter is split into four Laplace filters to obtain: （7）
（8）
（9）
（10）
and (3) operating all imaging results after RTM by using any modified Laplace filter shown in formulas (7), (8), (9) and (10), so as to denoising the seismic section.
Further, if the analysis is performed in the two-dimensional wave number domain, it is obtained by fourier transform:
（11）
in the method, in the process of the invention,kis the vector of the wave numbers of the imaging domain,k x is a reflection point edgexThe wave number of the direction is set,k z is a reflection point edgezWave number of direction;
FIG. 1 is a source wavefield vector of wavesSSum detector wave field vectorRSchematic representation and resulting imaging wavefield vectorn，θFor source wavefield vectorsSAnd imaging wave field vectorsRIs included in the bearing.
Wherein the imaging domain wavenumber vectorkExpressed as:
（12）
in the method, in the process of the invention,k R is the wave number vector of the wave field of the detector,k S is a source wave field wavenumber vector; applying the cosine law to obtain: （13）
wherein,vis the velocity of the seismic wave,ωis the angular frequency; from the above equation, the Laplace filter after fourier transform is an angle domain filter.
The wave field of the wave detector and the wave field of the seismic source shown in FIG. 2 are included with the wave field vector of the wave detector in an angle of 2θThe method comprises the steps of carrying out a first treatment on the surface of the In the conventional reverse time migration process, the included angles of all angles are imaged, and the drawing is performedAfter the Laplace filtering, the imaging result is added with the weight related to the angle, the weight is larger within 0-60 degrees, and the imaging result is well reserved; at the angle between the source wave field vector and the wave field vector of the wave detectorθWhen the angle is 60-90 degrees, the weight is smaller, so that the imaging wave field with a large angle, namely noise, is filtered.
From the above examples, the Laplace filter, which is an angle domain filter, has the following characteristics:
in the step 2), due toI 1 lap AndI 3 lap is symmetrical in mathematical sense, has smaller meaning for denoising the seismic section, and is preferably selected fromI 2 lap AndI 4 lap the resulting improved laplacian filter is,
will beI 2 lap Conversion to the wavenumber domain and analysis:
（14）
will beI 4 lap Conversion to the wavenumber domain and analysis:
（15）
in the method, in the process of the invention,αis the imaging wave field vectorzThe included angle of the directions is equivalent to adding a weighting factor cos to the wave number vector wave field after common Laplace filtering 2 αSum sin 2 αFor underground complex formations, the included angleαIs generally smaller, generally within the range of 0-30 DEG, the weighting factor cos 2 αCan well protect stratum imaging wave field vector, while sin 2 αThe formation imaging wave field vector can play a role in suppressing; on the other hand, the included angle of noiseαLarger weighting factor cos 2 αCan well suppress noise imaging wave field vectors, and correspondingly sin 2 αThe wave field vector of noise imaging can be actedProtection effect, so weight factor cos is selected 2 αI.e. the filter corresponding to equation (14), thus the LaplacianI 2 lap Further suppression of imaging noise may be performed.
Applying the improved Laplace filter reverse time offset method, namely four Laplace operators shown in the formulas (7) - (10), to the imaging results, and respectively filtering the image in the figure 3 to obtain the imaging results shown in the figures 4-7:
all Laplace operators suppress the low frequency noise generated by back reflection, but the response of each Laplace operator to the low frequency noise is different from that of each Laplace operatorxRelated to second-order differentiation of axial directionI 4 lap (as in FIG. 7) shows an offset image sensitive to the tilt configuration, whileI 1 lap AndI 3 lap (as in fig. 4 and 6) shows offset images that are sensitive to two opposite directional configurations, and also have severe crosstalk noise compared to the ideal offset results, causing destructive interference in the final stacked image. Related only to second-order differentiation in the z-axis directionI 2 lap The imaging effect is better for both the tilted configuration and the horizontal configuration (as shown in fig. 5), and the low frequency noise generated by back reflection is well suppressed.
The suppression of low frequency noise using the modified laplace operator relies on second order differential approximations. The modified Laplace operator is different from the Laplace operator composed of parallel items and normal items, and only the parallel items are selected to form the Laplace operator. We reverse time-shift the Fault model by cross-correlation imaging conditions (see fig. 3) using a modified laplace operator (see fig. 5). In comparison, low frequency noise suppression by back reflection is evident from the reverse time shift results using the modified Laplace operator.
Examples
Referring to fig. 8, the present example includes the steps of:
(1) The source wavefield is simulated based on the wave equation and boundary values are maintained at imaging time points.
(2) Obtaining a wave field of the detector by using wave equation reverse extension;
and (3) simulating the steps (1) and (2) by adopting a time second-order and space eighth-order finite difference wave equation, and normalizing the obtained seismic source wave field.
The Sigsbee2 model has a depth of 2.6km and a width of 7.6km (as shown in FIG. 9), and uses Rake wavelets with a dominant frequency of 30Hz and a time sampling interval of 1 ms. The shots are placed on the ground surface of 0 to 7500m, and the shot interval is set to 500m and 16 shots are taken as the whole. The detectors are uniformly arranged on the ground surface, the interval is 10m, the total number is 750, and the total recording length is 4s.
(3) The imaging condition of zero-delay cross-correlation is thatIn which, in the process,T max the maximum recording time is indicated and the time of the recording is indicated,SandRis a source wavefield and a detector wavefield.
(4) Setting Laplace filterIn the followingIRepresenting imaging result [ ]x,z) Representing the imaging point location.
(5) Respectively to Laplace operatorx、zThe second derivative in two directions is differentiated to obtain a discrete Laplace filter。
(6) Splitting eight neighborhood second-order differential Laplace operator, selecting Laplace operator related to horizontal second-order differential。
(7) Will beI 2 lap The denoising result obtained by applying the method to the RTM imaging result is shown in FIG. 10.
Claims (5)
1. The seismic section denoising method based on improved Laplace filtering reverse time migration is characterized by comprising the following steps of:
1) Collecting seismic data by using a detector;
2) Obtaining a source wave field and a detector wave field by using a reverse time migration method, and obtaining an imaging result after RTM processing by applying a zero-delay cross-correlation imaging condition:
the zero-delay cross-correlation imaging condition is expressed as
(one)
Wherein, the method comprises the following steps ofx,z) The position of the imaging point is indicated,T max the maximum recording time is indicated and the time of the recording is indicated,SandRis the source wavefield and the detector wavefield,I(x,z) Representing imaging results after RTM processing;
3) And (3) filtering:
suppressing low-frequency noise after imaging the imaging result by using a Laplace filter, wherein the Laplace filter is expressed as
(II)
In the middle ofI lap Representing imaging results after Laplace filteringx,z) The position of the imaging point is indicated,I(x,z) Representing the imaging result after RTM processing, laplacian 2 Is a second order differential operator in the Williams space, laplace operator is defined as
(III)
Carrying the formula (I) into the formula (II) to obtain an imaging result after Laplace filtering, wherein the imaging result is expressed as
(IV)
Discrete functionThe derivative of (a) is calculated in a differential mode, and the Laplace operator is respectively calculatedx、zLaplace operator filter for obtaining discrete function by differentiating second derivatives of two directions, which is expressed as
(V)
The eight-neighborhood second-order differential Laplace filter is split, and as the two opposite directions in the eight neighborhood have the same mathematical meaning, the eight-neighborhood second-order differential Laplace filter is split into four Laplace filters to obtain improved Laplace filtering:(six)
(seven)
(eight)
(nine)
And (3) operating the imaging result after the RTM processing by using any one improved Laplace filter shown in the formulas (six), (seven), (eight) and (nine), thereby denoising the seismic profile.
2. The improved laplacian filtered inverse time migration based seismic profile denoising method as claimed in claim 1, wherein in said step 2), the cross-correlation imaging condition is modified using normalization of the source wavefield, and the resulting imaging result is expressed as:
(ten).
3. The improved laplace filter reverse time migration based seismic profile denoising method as claimed in claim 1, wherein in said step 2), said RTM process comprises conventional acoustic wave RTM or elastic wave RTM.
4. The improved laplace filter reverse time migration based seismic profile denoising method as claimed in claim 1, wherein in said step 3), due toI 1 lap AndI 2 lap is symmetrical in mathematical sense, has smaller meaning for denoising the seismic section, and is preferably selected fromI 2 lap AndI 4 lap the improved laplacian filter is shown.
5. The method for denoising seismic profiles based on improved Laplace filtering reverse time migration of claim 4, wherein in step 3), the method is preferably selected from the group consisting ofI 2 lap The improved laplacian filter is shown.
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