CN112630835A - High-resolution post-stack seismic wave impedance inversion method - Google Patents
High-resolution post-stack seismic wave impedance inversion method Download PDFInfo
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- G01V1/28—Processing seismic data, e.g. analysis, for interpretation, for correction
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Abstract
The invention discloses a high-resolution post-stack seismic wave impedance inversion method, which comprises the following steps: constructing a forward model according to a time domain expression of the one-dimensional non-stationary seismic signal and an approximate expression of a reflection coefficient and wave impedance; establishing a longitudinal wave impedance low-frequency model by utilizing a relationship expression of longitudinal wave impedance and reflection coefficient and an approximate expression of the reflection coefficient and wave impedance; constructing the L with low-frequency constraint according to the Lagrange multiplier method, the forward model and the longitudinal wave impedance low-frequency model1‑2Minimizing the inverse objective function: l with low-frequency constraint and constructed by using convex function difference algorithm DCA and alternative method multiplier method ADMM pair1‑2And solving the minimum inversion target function to obtain longitudinal wave impedance.
Description
Technical Field
The invention relates to the technical field of post-stack seismic inversion, in particular to a high-resolution post-stack seismic wave impedance inversion method.
Background
The post-stack seismic inversion is to utilize the stacked seismic data to obtain reflection coefficient and wave impedance information through inversion, and focuses on describing reservoir characteristics. In recent 30 years, post-stack seismic inversion has made tremendous progress and is applied to reservoir prediction, reservoir characterization, and the like. In general, post-stack seismic inversion is divided into direct inversion based on reflection coefficient inverse formula and iterative inversion based on forward modeling, wherein the direct inversion method is most widely applied, and particularly, after Theune et al (2010) indicates that a block-shaped inversion result is favorable for depicting the boundary of the stratum, the inversion method is based on L1The regularization method and the improved method thereof are successfully applied to the post-stack seismic inversion. Wang et al uses L1The inversion of the seismic wave impedance is realized by the regularization and gradient descent method; zhang et al developed a seismic wave impedance inversion method based on fully-varying differentials, which respectively obtained the relative wave impedance and the absolute wave impedance; wanling et al have implemented the inversion of the relative wave impedance using the Toeplitz matrix sparse decomposition method with high accuracy, but the above methods all use a smoothly varying monopole wavelet base.
The steadily changing monopole wavelet base can cause low vertical resolution of inversion results. In order to solve the problem, Zhang et al (2011) construct a dipole wavelet base, and realize sparse inversion of a reflection coefficient by adopting basis pursuit, so that the method effectively improves the identification capability of a thin interbed; dawn et al (2015,2019) use a dipole wavelet base to perform the inversion of the relative and absolute wave impedances, respectively. In order to further improve the vertical resolution of the inversion result, Chai et al (2014) developed an inversion method of the nonstationary reflection coefficient based on a Margrave nonstationary convolution model; sui et al (2019) combine quality factor Q extraction and reflection coefficient inversion to achieve accurate reflection coefficient determination, but the Q used in the above method is fixed and does not change, and the Q actually changes with different features of the formation.
Disclosure of Invention
The invention provides a high-resolution post-stack seismic wave impedance inversion method, which solves the technical problem of how to improve the vertical resolution of inversion results of a weak reflecting layer and a deep reservoir.
The method for high-resolution post-stack seismic wave impedance inversion provided by the embodiment of the invention comprises the following steps:
constructing a forward model according to a time domain expression of the one-dimensional non-stationary seismic signal and an approximate expression of a reflection coefficient and wave impedance;
establishing a longitudinal wave impedance low-frequency model by utilizing a relationship expression of longitudinal wave impedance and reflection coefficient and an approximate expression of the reflection coefficient and wave impedance;
constructing the L with low-frequency constraint according to the Lagrange multiplier method, the forward model and the longitudinal wave impedance low-frequency model1-2Minimizing the inverse objective function:
l with low-frequency constraint and constructed by using convex function difference algorithm DCA and alternative method multiplier method ADMM pair1-2And solving the minimum inversion target function to obtain longitudinal wave impedance.
Preferably, the constructing a forward model according to the time domain expression of the one-dimensional non-stationary seismic signal and the approximate expression of the reflection coefficient and the wave impedance comprises:
obtaining a time domain expression of the one-dimensional non-stationary seismic signal according to the frequency domain expression of the one-dimensional non-stationary seismic signal:
according to the time domain expression of the one-dimensional non-stationary seismic signal and the approximate expression of the reflection coefficient and the wave impedance, a forward model is constructed, and the expression of the forward model is as follows: s ═ wa (q) Dm + n;
wherein ω represents angular frequency, τ represents time shift,fourier transform of w (t), r (t) is reflection coefficient, i is imaginary unit, t is sampling point, a (tau, omega) represents earthquake informationAmplitude attenuation of the sign, Q denotes the interlaminar Q value, W is the Toepliz matrix generated by wavelets, a (Q) denotes the viscoelastic attenuation matrix, m is the natural logarithm of the wave impedance, D is the first order difference matrix, s denotes the seismic record, and n denotes the noise sequence.
Preferably, the establishing a longitudinal wave impedance low-frequency model by using a relationship between longitudinal wave impedance and reflection coefficient and an approximate expression between reflection coefficient and wave impedance includes:
constructing a relational expression of a natural logarithmic sequence of the longitudinal wave impedance and a longitudinal wave impedance sequence according to the relational expression of the longitudinal wave impedance and the reflection coefficient and the approximate expression of the reflection coefficient and the wave impedance;
and establishing a longitudinal wave impedance low-frequency model according to the low-pass filtering matrix and the relation between the natural logarithm sequence of the longitudinal wave impedance and the longitudinal wave impedance sequence.
Preferably, the low-pass filter matrix comprises:
in the formula: n and m are rows and columns of corresponding elements in the low-pass filter matrix L; n is the length of the signal to be filtered; m is determined by the cut-off frequency, which is defined as:
in the formula: omegacIs the cut-off frequency.
Preferably, the longitudinal wave impedance low frequency model includes:
LCDm=ξlow
wherein C is an integral matrix and L is a low-frequency filter matrix.
Preferably, said L with low frequency constraint1-2Minimizing the inverse objective function includes:
in the formula, λ is a regularization parameter for adjusting the sparsity degree of the inversion result; the alpha is a weight parameter; beta is a weight coefficient of a longitudinal wave impedance low-frequency constraint term; and s is a seismic record.
Preferably, the L with low frequency constraint is constructed by using a convex function difference algorithm DCA and an alternative method multiplier method ADMM pair1-2Solving the minimum inversion target function to obtain longitudinal wave impedance comprises the following steps:
constraining the constructed L with low frequency1-2Carrying out first simplification processing on the minimum inversion target function to obtain L1-2Minimizing the first inversion objective function;
according to a DCA iterative formula, dividing the L1-2Minimizing the first inversion target function to perform second simplification processing to obtain L1-2Minimizing a second inversion objective function;
using the augmented Lagrange multiplier method for the L1-2And minimizing the second inversion target function for constraint to obtain a final inversion target function:
and obtaining the longitudinal wave impedance by using the final inversion objective function and the ADMM.
Preferably, the obtaining of the longitudinal wave impedance by using the final inversion objective function and the ADMM includes:
solving the final inversion target function by adopting ADMM to obtain a target inversion result:
in the formula, z and m are respectively updated and iterated by a soft threshold function and a gradient descent method, and solving formulas are respectively as follows:
zk+1=shrink(mk+wk/ρ,λ/ρ)
mk+1=(GTG+ρI)-1(GTd-yk+ρzk+1-wk)
in the formula, shrink is a soft threshold function, and when the iteration number of the final inversion target function reaches an upper limit, iteration is terminated to obtain a natural logarithm of longitudinal wave impedance;
and obtaining the longitudinal wave impedance according to the natural logarithm of the longitudinal wave impedance.
According to the scheme provided by the embodiment of the invention, the accuracy and the operation efficiency of the inversion result are improved, and the vertical resolution of the inversion result is improved.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the invention without limiting the invention. In the drawings:
FIG. 1 is a flow chart of a method for high resolution post-stack seismic wave impedance inversion according to an embodiment of the invention.
Detailed Description
The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, and it should be understood that the preferred embodiments described below are only for the purpose of illustrating and explaining the present invention, and are not to be construed as limiting the present invention.
Fig. 1 is a flowchart of a method for high-resolution post-stack seismic wave impedance inversion according to an embodiment of the present invention, as shown in fig. 1, including:
step S1: constructing a forward model according to a time domain expression of the one-dimensional non-stationary seismic signal and an approximate expression of a reflection coefficient and wave impedance;
step S2: establishing a longitudinal wave impedance low-frequency model by utilizing a relationship expression of longitudinal wave impedance and reflection coefficient and an approximate expression of the reflection coefficient and wave impedance;
step S3: constructing the L with low-frequency constraint according to the Lagrange multiplier method, the forward model and the longitudinal wave impedance low-frequency model1-2Minimizing the inverse objective function:
step S4: by means of projectionsL with low-frequency constraint and constructed by function difference algorithm DCA and alternative method multiplier method ADMM pair1-2And solving the minimum inversion target function to obtain longitudinal wave impedance.
Preferably, the constructing a forward model according to the time domain expression of the one-dimensional non-stationary seismic signal and the approximate expression of the reflection coefficient and the wave impedance comprises:
obtaining a time domain expression of the one-dimensional non-stationary seismic signal according to the frequency domain expression of the one-dimensional non-stationary seismic signal:
according to the time domain expression of the one-dimensional non-stationary seismic signal and the approximate expression of the reflection coefficient and the wave impedance, a forward model is constructed, and the expression of the forward model is as follows: s ═ wa (q) Dm + n;
wherein ω represents angular frequency, τ represents time shift,representing the Fourier transform of W (t), r (t) is the reflection coefficient, i is the imaginary unit, t is the sample point, a (τ, ω) represents the amplitude attenuation of the seismic signal, Q represents the interlaminar Q value, W is the Toepliz matrix generated by wavelets, A (Q) represents the viscoelastic attenuation matrix, m is the natural logarithm of the wave impedance, D is the first order difference matrix, s represents the seismic record, and n represents the noise sequence.
Preferably, the establishing a longitudinal wave impedance low-frequency model by using a relationship between longitudinal wave impedance and reflection coefficient and an approximate expression between reflection coefficient and wave impedance includes:
constructing a relational expression of a natural logarithmic sequence of the longitudinal wave impedance and a longitudinal wave impedance sequence according to the relational expression of the longitudinal wave impedance and the reflection coefficient and the approximate expression of the reflection coefficient and the wave impedance;
and establishing a longitudinal wave impedance low-frequency model according to the low-pass filtering matrix and the relation between the natural logarithm sequence of the longitudinal wave impedance and the longitudinal wave impedance sequence.
Preferably, the low-pass filter matrix comprises:
in the formula: n and m are rows and columns of corresponding elements in the low-pass filter matrix L; n is the length of the signal to be filtered; m is determined by the cut-off frequency, which is defined as:
in the formula: omegacIs the cut-off frequency.
Preferably, the longitudinal wave impedance low frequency model includes:
LCDm=ξlow
wherein C is an integral matrix and L is a low-frequency filter matrix.
Preferably, said L with low frequency constraint1-2Minimizing the inverse objective function includes:
in the formula, λ is a regularization parameter for adjusting the sparsity degree of the inversion result; the alpha is a weight parameter; beta is a weight coefficient of a longitudinal wave impedance low-frequency constraint term; and s is a seismic record.
Preferably, the L with low frequency constraint is constructed by using a convex function difference algorithm DCA and an alternative method multiplier method ADMM pair1-2Solving the minimum inversion target function to obtain longitudinal wave impedance comprises the following steps:
constraining the constructed L with low frequency1-2Carrying out first simplification processing on the minimum inversion target function to obtain L1-2Minimizing the first inversion objective function;
according to a DCA iterative formula, dividing the L1-2Minimizing the first inversion target function to perform second simplification processing to obtain L1-2Minimizing a second inversion objective function;
using the augmented Lagrange multiplier method for the L1-2And minimizing the second inversion target function for constraint to obtain a final inversion target function:
and obtaining the longitudinal wave impedance by using the final inversion objective function and the ADMM.
Preferably, the obtaining of the longitudinal wave impedance by using the final inversion objective function and the ADMM includes:
solving the final inversion target function by adopting ADMM to obtain a target inversion result:
in the formula, z and m are respectively updated and iterated by a soft threshold function and a gradient descent method, and solving formulas are respectively as follows:
zk+1=shrink(mk+wk/ρ,λ/ρ)
mk+1=(GTG+ρI)-1(GTd-yk+ρzk+1-wk)
in the formula, shrink is a soft threshold function, and when the iteration number of the final inversion target function reaches an upper limit, iteration is terminated to obtain a natural logarithm of longitudinal wave impedance;
and obtaining the longitudinal wave impedance according to the natural logarithm of the longitudinal wave impedance.
According to the method, the existing convex optimization methods such as basis tracking, total variation regularization and the like are distinguished, and a non-convex optimization algorithm is adopted to solve the inversion target function, so that the accuracy and the operational efficiency of the inversion result are improved, and the vertical resolution of the inversion result is improved.
The technical scheme of the application is specifically explained as follows:
1 establishing forward modeling
In Bickel and Natarajan (1985), a frequency domain expression of one-dimensional non-stationary seismic signals was derived, which is defined as:
where ω represents angular frequency, τ represents time shift,andrepresenting the Fourier transforms of s (t) and w (t), respectively. r (t) is the reflection coefficient, i is the imaginary unit, and t is the sampling point. a (τ, ω) represents the amplitude attenuation of the seismic signal, which is defined as:
in the formula, QeRepresents the equivalent Q value, which is defined as:
wherein Q represents an interlayer Q value. According to the inverse Fourier transform, a time domain expression of the one-dimensional non-stationary seismic signal can be obtained by the formula (1):
rewriting equation (4) into a convolution form can be obtained:
s(t)=w(t)*a(t,τ)·r(t)+n(t), (5)
where n (t) represents noise, and a (t, τ) represents the inverse fourier transform of a (τ, ω), which is defined as:
a(t,τ)=∫a(t,ω)eiωtdω. (6)
rewriting equation (5) to a matrix form can be obtained:
s=WA(Q)r+n, (7)
where W is the Toepliz matrix generated by the wavelets, s and r represent the seismic signal and reflection coefficient sequences, respectively, and n represents the noise sequence. A (Q) represents a viscoelastic damping matrix, which is represented by the following specific form:
according to the approximate expression of the reflection coefficient r (t) and the wave impedance I (t)The reflection coefficient can be defined as:
r=Dm, (9)
wherein m is the natural logarithm of the wave impedance, defined as mt(1/2) ln (i (t)). D is a first order difference matrix defined as:
and (3) combining the formula (7) and the formula (9) to obtain a final expression of the forward model:
s=WA(Q)Dm+n (10)
2 constructing a low frequency constraint term
At the sampling point corresponding to the time t, the relation between the longitudinal wave impedance and the reflection coefficient is as follows:
the form of the matrix equation, which is abbreviated as (11), is:
Cr=ξ (12)
in the formula: c is an integration matrix, and ξ is a relative wave impedance sequence, which is expressed as formula (13) and formula (14), respectively:
in combination with equation (9), equation (12) can be written as:
CDm=ξ (15)
equation (15) represents the relationship of the parameters to be inverted to the impedance sequence. In order to make the inversion result conform to the geological background of the actual work area, equation (15) is deformed as:
LCDm=ξlow (16)
in the formula, xilowThe low-frequency model is established according to the logging information, and L is a low-frequency filter matrix. In order to eliminate the effect, the patent designs a low-pass filter matrix suitable for seismic inversion according to DCT (Discrete Cosine Transform), and defines the low-pass filter matrix as follows:
in the formula: n and m represent the rows and columns of corresponding elements in the low-pass filter matrix L; n is the length of the signal to be filtered; m is determined by the cut-off frequency, which is defined as:
in the formula: omegacIs the cut-off frequency.
3 construction and solution of inversion objective function
According to the Lagrange multiplier method, the L with low-frequency constraint is obtained by combining the low-frequency constraint term formula (16)1-2Minimizing the inverse objective function:
in the formula, λ is a regularization parameter, the sparsity of the inversion result is adjusted, and α is a weight parameter, and β is a weight coefficient of the longitudinal wave impedance low-frequency constraint term.
equation (20) is decomposed into f (m) ═ g (m) -h (m) using DCA (Difference of covex Algorithm), where:
according to the DCA iterative formula, mixing L1-2The minimization of the inversion objective function is simplified as:
in the formula, ykIs H (m) at mkA gradient of (d), defined as:
the formula (22) is constrained by introducing an augmented Lagrange multiplier method, and the following can be obtained:
in the formula, z is an auxiliary intermediate variable, w is a Lagrange multiplier, and rho is a penalty parameter. Equation (24) is solved by an Alternative Direction Method of Multipliers (ADMM), and its iterative recursion equation is as follows:
in the formula, z and m are respectively updated and iterated by a soft threshold function and a gradient descent method, and solving formulas are respectively as follows:
zk+1=shrink(mk+wk/ρ,λ/ρ) (26)
mk+1=(GTG+ρI)-1(GTd-yk+ρzk+1-wk) (27)
in the formula, the shrink is a soft threshold function, when the iteration number of the formula (24) reaches the upper limit, the iteration is terminated, the natural logarithm m of the longitudinal wave impedance is obtained, and finally, the longitudinal wave impedance is obtained according to the formula (28).
Although the present invention has been described in detail hereinabove, the present invention is not limited thereto, and various modifications can be made by those skilled in the art in light of the principle of the present invention. Thus, modifications made in accordance with the principles of the present invention should be understood to fall within the scope of the present invention.
Claims (8)
1. A method of high resolution post-stack seismic wave impedance inversion, comprising:
constructing a forward model according to a time domain expression of the one-dimensional non-stationary seismic signal and an approximate expression of a reflection coefficient and wave impedance;
establishing a longitudinal wave impedance low-frequency model by utilizing a relationship expression of longitudinal wave impedance and reflection coefficient and an approximate expression of the reflection coefficient and wave impedance;
according to the Lagrange multiplier method, the forward model and the longitudinalWave impedance low frequency model, constructing L with low frequency constraint1-2Minimizing the inverse objective function:
l with low-frequency constraint and constructed by using convex function difference algorithm DCA and alternative method multiplier method ADMM pair1-2And solving the minimum inversion target function to obtain longitudinal wave impedance.
2. The method of claim 1, wherein constructing the forward model based on the time domain representation of the one-dimensional non-stationary seismic signals and the approximate representation of reflection coefficients and wave impedance comprises:
obtaining a time domain expression of the one-dimensional non-stationary seismic signal according to the frequency domain expression of the one-dimensional non-stationary seismic signal:
according to the time domain expression of the one-dimensional non-stationary seismic signal and the approximate expression of the reflection coefficient and the wave impedance, a forward model is constructed, and the expression of the forward model is as follows: s ═ wa (q) Dm + n;
wherein ω represents angular frequency, τ represents time shift,representing the Fourier transform of W (t), r (t) is the reflection coefficient, i is the imaginary unit, t is the sample point, a (τ, ω) represents the amplitude attenuation of the seismic signal, Q represents the interlaminar Q value, W is the Toepliz matrix generated by wavelets, A (Q) represents the viscoelastic attenuation matrix, m is the natural logarithm of the wave impedance, D is the first order difference matrix, s represents the seismic record, and n represents the noise sequence.
3. The method of claim 2, wherein the establishing a low frequency model of the longitudinal wave impedance using the relationship between the longitudinal wave impedance and the reflection coefficient and the approximate expression of the reflection coefficient and the wave impedance comprises:
constructing a relational expression of a natural logarithmic sequence of the longitudinal wave impedance and a longitudinal wave impedance sequence according to the relational expression of the longitudinal wave impedance and the reflection coefficient and the approximate expression of the reflection coefficient and the wave impedance;
and establishing a longitudinal wave impedance low-frequency model according to the low-pass filtering matrix and the relation between the natural logarithm sequence of the longitudinal wave impedance and the longitudinal wave impedance sequence.
4. The method of claim 3, wherein the low pass filter matrix comprises:
in the formula: n and m are rows and columns of corresponding elements in the low-pass filter matrix L; n is the length of the signal to be filtered; m is determined by the cut-off frequency, which is defined as:
in the formula: omegacIs the cut-off frequency.
5. The method of claim 4, wherein the longitudinal wave impedance low frequency model comprises:
LCDm=ξlow
wherein C is an integral matrix and L is a low-frequency filter matrix.
6. The method of claim 5, wherein the L with low frequency constraint1-2Minimizing the inverse objective function includes:
in the formula, λ is a regularization parameter for adjusting the sparsity degree of the inversion result; the alpha is a weight parameter; beta is a weight coefficient of a longitudinal wave impedance low-frequency constraint term; and s is a seismic record.
7. The method according to claim 6, wherein the L with low frequency constraint is constructed by using a convex function difference algorithm DCA and an alternative method multiplier method ADMM pair1-2Solving the minimum inversion target function to obtain longitudinal wave impedance comprises the following steps:
constraining the constructed L with low frequency1-2Carrying out first simplification processing on the minimum inversion target function to obtain L1-2Minimizing the first inversion objective function;
according to a DCA iterative formula, dividing the L1-2Minimizing the first inversion target function to perform second simplification processing to obtain L1-2Minimizing a second inversion objective function;
using the augmented Lagrange multiplier method for the L1-2And minimizing the second inversion target function for constraint to obtain a final inversion target function:
and obtaining the longitudinal wave impedance by using the final inversion objective function and the ADMM.
8. The method of claim 7, wherein obtaining longitudinal wave impedance using the final inverted objective function and the ADMM comprises:
solving the final inversion target function by adopting ADMM to obtain a target inversion result:
in the formula, z and m are respectively updated and iterated by a soft threshold function and a gradient descent method, and solving formulas are respectively as follows:
zk+1=shrink(mk+wk/ρ,λ/ρ)
mk+1=(GTG+ρI)-1(GTd-yk+ρzk+1-wk)
in the formula, shrink is a soft threshold function, and when the iteration number of the final inversion target function reaches an upper limit, iteration is terminated to obtain a natural logarithm of longitudinal wave impedance;
and obtaining the longitudinal wave impedance according to the natural logarithm of the longitudinal wave impedance.
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