CN116068621A - Anisotropic medium forward modeling method and system based on rigidity matrix decomposition - Google Patents
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Abstract
The invention discloses an anisotropic medium forward modeling method and system based on rigidity matrix decomposition, wherein the method comprises the following steps: splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation; deducing a first-order velocity-stress equation of longitudinal wave separation of the anisotropic medium according to the longitudinal wave equation and the transverse wave equation; and performing anisotropic medium forward modeling according to the first-order speed-stress equation. According to the method, the elastic wave equation is split into the longitudinal wave equation and the transverse wave equation, and the first-order speed-stress equation of longitudinal and transverse wave separation facing the anisotropic medium is deduced according to the split longitudinal wave equation and transverse wave equation, so that the amplitude and phase information of a longitudinal and transverse wave field obtained by the method are more accurate, and the wave field information is more complete.
Description
Technical Field
The invention belongs to the field of petroleum geological exploration, and particularly relates to an anisotropic medium forward modeling method and system based on rigidity matrix decomposition.
Background
Anisotropy exists widely in the earth medium, and the anisotropic medium has the characteristic of longitudinal and transverse wave coupling, so that the forward modeling method based on isotropic assumption is insufficient for accurately describing the propagation rule of the seismic waves in the actual earth medium. Although anisotropic elastic wave equation forward modeling has made some progress, the following drawbacks also exist: firstly, the anisotropic parameters are complex, the physical meaning is undefined, and the anisotropic parameters are difficult to obtain in actual production; secondly, because the cost of multi-component seismic exploration is higher, the current field acquisition still takes longitudinal wave seismic data as the main material, and the multi-component elastic wave theory is limited by insufficient data. Therefore, the longitudinal and transverse waves of the anisotropic medium need to be separated so as to meet the requirements of seismic data acquisition and processing at the current stage. At present, two types of methods are mainly used for extracting longitudinal wave components in an anisotropic medium, namely, a longitudinal wave field separation method based on longitudinal wave polarization direction difference is generally adopted, a Helmholtz decomposition method is generally adopted, zhu ([ 1]Zhu,Hejun.Elastic wavefield separation based on the Helmholtz decomposition[J ]. Geophysics: journal of the Society of Exploration Geophysicists,2017,82 (2): S173-S183.), and a pseudo-acoustic wave (qP wave) approximation equation is deduced through a dispersion relation.
The propagation direction of the seismic wave in the anisotropic medium is different from that of the isotropic medium, and the relationship that the vibration direction of the longitudinal wave is the same as the propagation direction and the vibration direction of the transverse wave is perpendicular to the wave field propagation direction is not adopted. Therefore, the helmholtz decomposition method based on the difference in polarization directions of the longitudinal and transverse waves is not very effective for achieving longitudinal and transverse wave separation. In addition, in the separation process, a divergence field is used for representing a longitudinal wave, a rotation field is used for representing a transverse wave, so that a vector longitudinal wave field is changed into a scalar wave field, the physical meaning of the vector longitudinal wave field is different from that of an original wave field, and the amplitude and phase information are changed.
In terms of the qP wave approximation equation, alkhalifah (alkhalifaht. Objective approximations for processing in transversely isotropic media. Geodynamics, 1998,63 (2): 623-631;Alkhalifah T.An acoustic wave equation for anisotropic media.Geophysics,2000,65 (4): 1239-1250.) derives the VTI medium fourth order pseudo-acoustic equation, starting from the exact qP-qSV wave dispersion relationship of the VTI medium, assuming a vertical shear wave velocity Vs0 = 0. To reduce the complexity of the fourth order partial differential equation calculation. Two VTI medium second order coupling pseudo-acoustic wave equations are derived from acoustically approximated dispersion relations by introducing different auxiliary wavefield functions, two types of VTI medium second order coupling pseudo-acoustic wave equations are derived, respectively, by Zhou (Zhou H, zhang G, zhang Y.an anisotropic acoustic wave equation for VTI medium.68 th EAGE Conference and Exhibit.Vienna: european Association of Geoscientists and Engineers,2006: 1-5.) and Du (Du X, fletcher R, fowler P J.A new pseudo-acoustic wave equation for VTI medium.70 th EAGE Conference and Exhibit.Rome: european Association of Geoscientists and Engineers,2008: 1-5.). To achieve accurate finite difference modeling in combination with staggered grid techniques, hestholm (Hestholm S. Objective VTI modeling using higher-order fine differences, 2009,74 (5): T67-T73.) derives a first order partial differential form of the VTI dielectric pseudo-acoustic equation. Duveneck et al (Duveneck E, milcik P, bakker P m. Operational VTI wave equations and their application for anisotropic reverse-time division. Seg Technical Program Expanded abstracts. Las Vegas: society of Exploration Geophysicists, 2008:2186-2190.) also derive VTI media pseudo-acoustic equations in second order form and first order velocity-stress form, starting from the acoustical approximation hooke's law. Similar to the method of Alkhalifah, several pseudo-acoustic equations have the problem of low-speed, low-amplitude qSV artificial interference waves (Grechka V., zhang L, and J.W. Rector,2004, shear-waves in acoustic anisotropic media, geophysics,69 (2): 576-582; jin S and A. Stovas,2018, S-wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis, geophysical Prospecting,66 (6): 1123-1137), which affect the end result of forward modeling and offset imaging. In order to solve the problem of low-speed and low-amplitude qSV artificial interference waves in the traditional qP wave approximation method, xu et al (Xu S.B., stovas A, alkhalifah T, and Mikada H, 2020,New acoustic approximation for transversely isotropic media with a vertical symmetry axis,Geophysics,85 (1): C1-C12) proposes a novel qP wave approximation method, which is based on the fact that the vertical SV wave velocity is not assumed to be zero, but the SV wave phase velocity is made to be zero, so that the vertical SV wave velocity expression is obtained under the condition, and the qP wave phase velocity is calculated. However, the difference between the qP wave phase velocity calculated by this method and the phase velocity calculated by the Alkhalifah (Alkhalifah T.an acoustic wave equation for anisotropic media. Geophysics,2000,65 (4): 1239-1250.) method is small (as shown in FIG. 1), and if the vertical SV wave velocity expression is substituted into the SV wave phase velocity expression, the resulting SV wave phase velocity is not zero. In the method proposed by Xu et al (Xu s.b., stovas a., alkhalifah T, and Mikada h.,2020,New acoustic approximation for transversely isotropic media with a vertical symmetry axis,Geophysics,85 (1): C1-C12.), only the qP wave phase velocity expression is given, and no wave field forward simulation result is given. Another disadvantage of the qP wave approximation method is that the resulting longitudinal wave data of the anisotropic medium is incomplete because the information of the converted wave cannot be taken into account.
The change in qP wave phase velocity with polarization angle calculated according to the Xu et al (2020) and Alkhalifah (2000) methods is shown in FIG. 1. Model parameters are the same as model 1 (Vp 0=3 km/s, vs0=1 km/s, vpn=3.3 km/s, anisotropy parameter η=0.1) in Xu et al (2020).
Disclosure of Invention
In order to solve the problems, the invention provides an anisotropic medium forward modeling method based on rigidity matrix decomposition, which comprises the following steps:
splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation;
deducing a first-order velocity-stress equation of longitudinal wave separation of the anisotropic medium according to the longitudinal wave equation and the transverse wave equation;
and performing anisotropic medium forward modeling according to the first-order speed-stress equation.
Further, the splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation comprises the following steps:
each coefficient C in the rigidity coefficient matrix C 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 Decomposing into a linear function of longitudinal wave modulus and/or transverse wave modulus; wherein the stiffness matrix C is in the form of:
among the above coefficients, C 11 ＝C 22 ，C 13 ＝C 23 And C 44 ＝C 55 Equal coefficients are represented in the stiffness matrix C by only one of the coefficients.
Respectively establishing a rigidity coefficient matrix C related to longitudinal waves according to the linear function p Transverse wave related stiffness coefficient matrix C s ；
According to the C p And C s Wave equation of second-order elastic wave
Splitting into longitudinal wave equation and transverse wave equation.
Further, each coefficient C in the stiffness coefficient matrix C 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 The expression of (2) is as follows:
wherein epsilon is a parameter for measuring the anisotropic strength of the qS wave; delta is the velocity V of the longitudinal wave connecting in the direction of the symmetry axis p0 And longitudinal wave velocity V in the direction perpendicular to the axis of symmetry p90 A transition parameter therebetween; gamma is a parameter for measuring the anisotropic intensity of the qS wave or the transverse wave splitting intensity;
is longitudinal wave transverse quantity +.>
For transverse wave transverse quantity, ρ represents density.
Further, the longitudinal wave equation is:
the transverse wave equation is:
wherein C is p C is a matrix of stiffness coefficients related to longitudinal waves s As a matrix of elastic coefficients related to transverse waves, u p As a longitudinal wave field, u s As a transverse wave field, a full wave field u=u p +u s ρ represents density, t represents time, and G is in the form of a gradient matrix:
wherein x, y, z are spatial coordinates.
Further, the method comprises the steps of,
the said
Wherein C is p Is a rigidity coefficient matrix related to longitudinal waves; c (C) s Is an elastic coefficient matrix related to transverse waves; epsilon is a parameter for measuring the strength of anisotropy of the qS wave; delta is the velocity V of the longitudinal wave connecting in the direction of the symmetry axis p0 And longitudinal wave velocity V in the direction perpendicular to the axis of symmetry p90 A transition parameter therebetween; gamma is a parameter for measuring the anisotropic intensity of the qS wave or the transverse wave splitting intensity;
is longitudinal wave transverse quantity +.>
For transverse wave transverse quantity, ρ represents density.
Further, the first-order velocity-stress equation is:
wherein V is P Representing longitudinal wave velocity vector, V S Represents a transverse wave velocity vector, t represents time, v=v P +V S Representing the total velocity vector, C p Is a rigidity coefficient matrix related to longitudinal waves; c (C) s Is an elastic coefficient matrix related to transverse waves; ρ represents density, G represents gradient matrix, C represents stiffness coefficient matrix, and T represents stress vector;
Τ＝(σ xx σ yy σ zz σ yz σ xz σ xy ) T
wherein sigma xx 、σ yy Sum sigma zz Respectively represents positive stress and sigma of the main axis in the x, y and z directions yz 、σ xz Sum sigma xy Denote the shear stress in yz, xz and xy planes, respectively, and the superscript T denotes the vector transposition.
Further, the first-order velocity-stress equation is derived as follows:
calculation was performed according to hooke's law:
Τ＝CE＝CG T u (14)
wherein T is a stress vector, C is a rigidity coefficient matrix, E is a strain vector, and u represents a full wave field;
E＝(e xx e yy e zz e yz e xz e xy ) T
wherein e xx 、e yy 、e zz Representing positive strain in the x, y, z direction, e yz 、e xz 、e xy Representing the tangential strains in the yz, xz and xy planes, the superscript T representing the vector transpose;
as can be seen from the formula (14),
G T u＝C -1 Τ (15)
let the velocity vector
Wherein u is p Representing longitudinal wave displacement vector, u s Represents a transverse wave displacement vector according to the formula G T u＝C -1 And converting the sum of the longitudinal wave equation and the transverse wave equation into a first-order velocity-stress equation.
The invention also provides an anisotropic medium forward modeling system based on rigidity matrix decomposition, which comprises the following modules:
the splitting module is used for splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation;
the deduction module is used for deducting a first-order speed-stress equation of longitudinal wave separation of the anisotropic medium according to the longitudinal wave equation and the transverse wave equation;
and the forward modeling module is used for carrying out anisotropic medium forward modeling according to the first-order speed-stress equation.
Further, the splitting module includes:
a decomposition unit for decomposing each coefficient C in the rigidity coefficient matrix C 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 Decomposing into a linear function of longitudinal wave modulus and/or transverse wave modulus;
the matrix building unit is used for respectively building a rigidity coefficient matrix related to longitudinal waves and a rigidity coefficient matrix related to transverse waves according to the linear function;
a splitting unit for splitting the second-order elastic wave equation
Splitting into longitudinal wave equation and transverse wave equation.
Further, the longitudinal wave equation is:
the transverse wave equation is:
wherein C is p C is a matrix of stiffness coefficients related to longitudinal waves s As a matrix of elastic coefficients related to transverse waves, u p As a longitudinal wave field, u s As a transverse wave field, a full wave field u=u p +u s ρ represents density and t represents time, where G is a gradient matrix in the form of:
wherein x, y, z are space coordinates;
the first order speed-stress equation is:
wherein V is P Representing longitudinal wave velocity vector, V S Represents a transverse wave velocity vector, t represents time, v=v P +V S Representing the total velocity vector, ρ representing density, G representing the gradient matrix, and T representing the stress vector.
The invention has the beneficial effects that:
according to the method, the elastic wave equation is split into the longitudinal wave equation and the transverse wave equation, and the first-order speed-stress equation of longitudinal and transverse wave separation facing the anisotropic medium is deduced according to the split longitudinal wave equation and transverse wave equation, so that the amplitude and phase information of the longitudinal and transverse wave field obtained by the method are more accurate, the wave field information is more complete, and the anisotropic medium forward modeling can be better performed.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The objectives and other advantages of the invention may be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions of the prior art, the following description will briefly explain the drawings used in the embodiments or the description of the prior art, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings can be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 shows a graph of the calculated qP wave phase velocity as a function of polarization angle according to the methods of Xu et al (2020) and Alkhalifah (2000) in the background;
FIG. 2 shows a forward model employed in an embodiment of the present invention;
FIG. 3 shows an elastic wave field diagram obtained by a conventional anisotropic elastic wave forward method in an embodiment of the present invention;
FIG. 4 is a graph showing Y and X components of a longitudinal wave field and a transverse wave field obtained by a Helmholtz decomposition method according to an embodiment of the present invention;
FIG. 5 shows a qP wave forward result obtained by adopting a pseudo-acoustic wave approximation method in an embodiment of the invention;
FIG. 6 shows a longitudinal wave field diagram obtained by adopting the anisotropic medium forward method based on rigidity coefficient matrix decomposition provided by the invention in the embodiment of the invention;
fig. 7 shows a transverse wave field diagram obtained by adopting the anisotropic medium forward method based on stiffness coefficient matrix decomposition according to the embodiment of the invention.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In order to obtain anisotropic medium longitudinal wave data with relatively accurate amplitude and phase information and relatively complete wave field information, the invention obtains a first-order speed-stress equation of anisotropic medium longitudinal wave separation by approximating a coupling term in a rigidity coefficient matrix as a linear function of longitudinal wave modulus and transverse wave modulus, thereby realizing the longitudinal wave field separation of the anisotropic medium. Compared with other wave field separation algorithms, the amplitude and phase information of the longitudinal and transverse wave fields obtained based on the equation separation algorithm is more accurate, and the wave field information is more complete.
The invention provides an anisotropic medium forward modeling method based on rigidity matrix decomposition, which comprises the following steps:
step one: splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation;
the wave equation is expressed by a stiffness matrix C, and after the stiffness matrix C is decomposed into a longitudinal wave component and a transverse wave component, the wave equation can be decomposed into a longitudinal wave equation and a transverse wave equation. Decomposing the rigidity coefficient matrix C into linear functions of longitudinal wave modulus and transverse wave modulus; specifically, the stiffness coefficient matrix C may be decomposed into a portion related to a longitudinal wave and a portion related to a transverse wave, so that the elastic wave equation is split into a longitudinal wave equation and a transverse wave equation.
Wherein, the form of rigidity coefficient matrix is:
wherein C is 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 Representing the individual coefficients of the stiffness matrix.
Among the above coefficients, C 11 ＝C 22 ，C 13 ＝C 23 And C 44 ＝C 55 Equal coefficients are represented in the stiffness matrix C by only one of the coefficients.
In the embodiment of the invention, a splitting process is described by taking a VTI medium as an example.
For VTI media, components with non-zero stiffness coefficients are represented by Thomsen parameters as:
wherein C is 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 Are the individual coefficients in the stiffness matrix C in equation (1).
The formulae (2) - (7) are written in a matrix form as shown in formula (8):
wherein V is p0 And V s0 The phase velocities of the qP wave and qSV wave along the symmetry axis of the VTI medium are three dimensionless factors representing the strength of the VTI medium anisotropy. Wherein epsilon is a parameter for measuring the anisotropic strength of the qS wave, and the larger epsilon is, the larger epsilon is the anisotropic strength of the longitudinal wave of the medium; delta is the velocity V of the longitudinal wave connecting in the direction of the symmetry axis p0 And longitudinal wave velocity V in the direction perpendicular to the axis of symmetry p90 A transition parameter therebetween; gamma can be regarded as a parameter measuring the intensity of the qS wave anisotropy or the intensity of the shear wave splitting, and the larger gamma is, the larger the shear wave anisotropy of the medium is, and ρ represents the density.
As can be seen from formulas (2) - (7), except C 13 And C 23 In addition, the coefficients are all longitudinal wave moduli
And transverse wave modulus->
Is a linear function of (c). If C is to be 13 And C 23 Expressed as longitudinal wave modulus +.>
And transverse wave modulus->
The stiffness coefficient matrix C can be decomposed into longitudinal wave related parts C p And a section C related to transverse waves s 。
Splitting the Cheng Zongbo equation and the transverse wave equation for the elastic wave equation in the first step comprises the following steps:
1. decomposing the rigidity coefficient matrix C into linear functions of longitudinal wave modulus and transverse wave modulus;
due to the stiffness matrix coefficient C 13 And C 23 Not longitudinal wave modulus
And/or transverse wave modulus->
Thus, can be applied to C 13 And C 23 The modification of the expression (5) to obtain the expression (9)
In the formula (9), the amino acid sequence of the compound,
in most parameter configuration cases, |Δ| < 1, due to
The maximum value of (2) is 0.707, thus +.>
Has a maximum value of about 0.25, and +.>
Is small in comparison, thus +.>
This term is truncated from the formula Δ, which can be converted to:
taylor expansion of (11)
Whereby the stiffness coefficient C can be determined 13 And C 23 Conversion to longitudinal wave modulus
And transverse wave modulus->
Is a linear function of (c).
2. Respectively establishing a rigidity coefficient matrix C related to longitudinal waves according to the linear function p Transverse wave related stiffness coefficient matrix C s 。
Longitudinal wave related stiffness coefficient matrix C in VTI medium p Transverse wave related stiffness coefficient matrix C s The method comprises the following steps of:
3. thereby, the second-order elastic wave equation can be used
The equations are divided into longitudinal wave equations and transverse wave equations in the equations (15) - (16).
The split longitudinal wave equation and transverse wave equation are as follows:
the longitudinal wave equation is:
the transverse wave equation is:
wherein C is p C is a matrix of stiffness coefficients related to longitudinal waves s As a matrix of elastic coefficients related to transverse waves, u p And u s Longitudinal wave field and transverse wave field, respectively, full wave field u=u p +u s ρ represents density, T represents time, the superscript T represents vector transpose, G is gradient matrix, in the form:
wherein x, y, z are spatial coordinates.
Step two: and deducing a first-order speed-stress equation of longitudinal and transverse wave separation facing the anisotropic medium according to the split longitudinal wave equation and transverse wave equation.
In order to facilitate the forward modeling of the earthquake by using the staggered grid method, the longitudinal wave equation and the transverse wave equation in the formulas (15) and (16) are converted into the form of a first-order velocity-stress equation.
The first order velocity-stress equation is obtained as follows:
calculated according to Hooke's law, the expression of Hooke's law is
Τ＝CE＝CG T u (18)
Wherein, T= [ sigma ] xx σ yy σ zz σ yz σ xz σ xy ] T Is the stress vector, C is the stiffness coefficient matrix, e= (E) xx e yy e zz e yz e xz e xy ) T Is a strain vector; wherein sigma xx 、σ yy 、σ zz Respectively represents positive stress and sigma of the main axis in the x, y and z directions yz 、σ xz Sum sigma xy Representing the shear stress in yz, xz and xy planes, respectively, the superscript T represents the vector transpose, where e xx 、e yy 、e zz Representing positive strain in the x, y, z direction, e yz 、e xz 、e xy Representing the tangential strains in the yz, xz and xy planes, the superscript T representing the vector transpose, and u representing the full wavefield;
as can be seen from the formula (18),
G T u＝C -1 Τ (19)
let the velocity vector
By the formula (19), u p And u s Longitudinal wave field and transverse wave field, respectively, full wave field u=u p +u s . Equation (15) and equation (16) can be converted into a first order velocity-stress equation of the form:
wherein V is P Representing longitudinal wave velocity vector, V S Represents a transverse wave velocity vector, t represents time, v=v P +V S Representing the total velocity vector, ρ representing density, G representing the gradient matrix, and T representing the stress vector.
Examples
The anisotropic medium forward modeling method based on the rigidity matrix decomposition is tested. The method is tested by adopting a three-layer horizontal lamellar medium model, and model parameters comprise five rigidity coefficients C 11 ，C 33 ，C 13 ，C 44 ，C 66 And density. A three-dimensional model is adopted, the size of the model is 2400m (x) x 2370m (y) x 2700m (z), the size of the space grid in the directions of x, y and z is 15m, and the model is shown in figure 2.
The conditions for the simulation were: the physical time of the simulation was 1s, the time step was 0.5ms, the primary frequency was 30Hz with a rake wavelet source located at x=1350 m, y=1185 m, z=465 m.
The model of fig. 2 is subjected to conventional anisotropic elastic wave forward modeling by using the simulation conditions, the obtained wave field is shown in fig. 3, the wave field obtained on each component in fig. 3 comprises longitudinal wave data and transverse wave data from left to right, wherein the wave field is respectively a Z component, a Y component and an X component in fig. 3. Since the impedance difference between the third layer and the second layer is small, the corresponding reflected signal is weak and hardly visible in the figure.
The mixed wave field in fig. 2 was subjected to longitudinal and transverse wave separation by using the helmholtz decomposition method, and the obtained results are shown in fig. 4. Since the longitudinal wave field obtained by the helmholtz decomposition method is a scalar field, there is only one component. The Y and X components of the separated transverse wave wavefield are shown simultaneously in fig. 4. Comparing fig. 4 and fig. 3, it can be seen that the longitudinal wave and transverse wave data obtained by the helmholtz decomposition method are distorted in amplitude and phase compared to the original wave field.
The three components of the obtained pseudo-acoustic wave are shown in fig. 5, namely a Z component, a Y component and an X component from left to right in the figure, by using the pseudo-acoustic wave approximate forward modeling method of Alkhalifar (2000). As can be seen from fig. 5, SV interference waves exist in longitudinal wave data obtained by a forward method based on pseudo-acoustic approximation (S1 in the figure). In addition, as can be seen by comparing with the longitudinal wave field in fig. 3, the longitudinal wave data obtained by the pseudo-acoustic approximation method is different in both amplitude and phase from the real longitudinal wave, and the pseudo-acoustic approximation method does not obtain converted longitudinal wave information (P2) in the original wave field.
The anisotropic medium forward modeling method based on the rigidity coefficient matrix decomposition is applied to the parameter model in fig. 2, and the obtained longitudinal wave field data and transverse wave field data are respectively shown in fig. 6 and 7, and are respectively Z component, Y component and X component from left to right in fig. 6 and 7. As can be seen by comparing the forward result of the invention with the forward result of the Helmholtz decomposition method and the quasi-acoustic wave approximation method, the anisotropic medium forward method based on the rigidity coefficient matrix decomposition can obtain longitudinal wave data and transverse wave data with accurate amplitude and phase information.
From the comparison results, it can be seen that the anisotropic medium forward modeling method based on the stiffness coefficient matrix decomposition provided by the invention can be applied to the parameter model in fig. 2 to obtain more accurate amplitude and phase information of the longitudinal wave field and wave field information.
Although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims (10)
1. An anisotropic medium forward modeling method based on rigidity matrix decomposition is characterized by comprising the following steps:
splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation;
deducing a first-order velocity-stress equation of longitudinal wave separation of the anisotropic medium according to the longitudinal wave equation and the transverse wave equation;
and performing anisotropic medium forward modeling according to the first-order speed-stress equation.
2. The anisotropic media forward method based on stiffness matrix factorization of claim 1,
the splitting of the elastic wave equation into a longitudinal wave equation and a transverse wave equation comprises the following steps:
each coefficient C in the rigidity coefficient matrix C 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 Decomposing into a linear function of longitudinal wave modulus and/or transverse wave modulus; wherein the stiffness matrix C is in the form of:
respectively establishing a rigidity coefficient matrix C related to longitudinal waves according to the linear function p Transverse wave related stiffness coefficient matrix C s ；
According to the C p And C s Wave equation of second-order elastic wave
Splitting into longitudinal wave equation and transverse wave equation.
3. The anisotropic media forward method based on stiffness matrix factorization of claim 2,
each coefficient C in the stiffness coefficient matrix C 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 The expression of (2) is as follows:
wherein epsilon is a parameter for measuring the anisotropic strength of the qS wave; delta is the velocity V of the longitudinal wave connecting in the direction of the symmetry axis p0 And longitudinal wave velocity V in the direction perpendicular to the axis of symmetry p90 A transition parameter therebetween; gamma is a parameter for measuring the anisotropic intensity of the qS wave or the transverse wave splitting intensity;
is longitudinal wave transverse quantity +.>
For transverse wave transverse quantity, ρ represents density.
4. The anisotropic medium forward method based on stiffness matrix decomposition according to claim 1 or 2, wherein,
the longitudinal wave equation is:
the transverse wave equation is:
wherein C is p C is a matrix of stiffness coefficients related to longitudinal waves s As a matrix of elastic coefficients related to transverse waves, u p As a longitudinal wave field, u s As a transverse wave field, a full wave field u=u p +u s ρ represents density, t represents time, and G is in the form of a gradient matrix:
wherein x, y, z are spatial coordinates.
5. The anisotropic media forward method based on stiffness matrix factorization of claim 4,
the said
Wherein C is p Is a rigidity coefficient matrix related to longitudinal waves; c (C) s Is an elastic coefficient matrix related to transverse waves; epsilon is a parameter for measuring the strength of anisotropy of the qS wave; delta is the velocity V of the longitudinal wave connecting in the direction of the symmetry axis p0 And longitudinal wave velocity V in the direction perpendicular to the axis of symmetry p90 A transition parameter therebetween; gamma is a parameter for measuring the anisotropic intensity of the qS wave or the transverse wave splitting intensity;
is longitudinal wave transverse quantity +.>
For transverse wave transverse quantity, ρ represents density.
6. The anisotropic media forward method based on stiffness matrix factorization of claim 1,
the first order speed-stress equation is:
wherein V is P Representing longitudinal wave velocity vector, V S Representing transverseWave velocity vector, t represents time, v=v P +V S Representing the total velocity vector, C p Is a rigidity coefficient matrix related to longitudinal waves; c (C) s Is an elastic coefficient matrix related to transverse waves; ρ represents density, G represents gradient matrix, C represents stiffness coefficient matrix, and T represents stress vector;
Τ＝(σ xx σ yy σ zz σ yz σ xz σ xy ) T
wherein sigma xx 、σ yy Sum sigma zz Respectively represents positive stress and sigma of the main axis in the x, y and z directions yz 、σ xz Sum sigma xy Denote the shear stress in yz, xz and xy planes, respectively, and the superscript T denotes the vector transposition.
7. The anisotropic media forward method based on stiffness matrix decomposition according to claim 1 or 6, wherein,
the first order velocity-stress equation is derived as follows:
calculation was performed according to hooke's law:
Τ＝CE＝CG T u (14)
wherein T is a stress vector, C is a rigidity coefficient matrix, E is a strain vector, and u represents a full wave field;
E＝(e xx e yy e zz e yz e xz e xy ) T
wherein e xx 、e yy 、e zz Representing positive strain in the x, y, z direction, e yz 、e xz 、e xy Representing the tangential strains in the yz, xz and xy planes, the superscript T representing the vector transpose;
as can be seen from the formula (14),
G T u＝C -1 Τ (15)
let the velocity vector
Wherein u is p Representing longitudinal wave displacement vector, u s Representing transverse wave positionMotion vector according to the formula G T u＝C -1 And converting the sum of the longitudinal wave equation and the transverse wave equation into a first-order velocity-stress equation.
8. An anisotropic medium forward system based on rigidity matrix decomposition, which is characterized by comprising the following modules:
the splitting module is used for splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation;
the deduction module is used for deducting a first-order speed-stress equation of longitudinal wave separation of the anisotropic medium according to the longitudinal wave equation and the transverse wave equation;
and the forward modeling module is used for carrying out anisotropic medium forward modeling according to the first-order speed-stress equation.
9. The anisotropic media forward system based on stiffness matrix factorization of claim 8,
the splitting module comprises:
a decomposition unit for decomposing each coefficient C of the rigidity coefficients C 11 ，C 22 ，C 33 ，C 12 ，C 13 ，C 23 ，C 44 ，C 55 ，C 66 Decomposing into a linear function of longitudinal wave modulus and/or transverse wave modulus;
the matrix building unit is used for respectively building a rigidity coefficient matrix related to longitudinal waves and a rigidity coefficient matrix related to transverse waves according to the linear function;
a splitting unit for splitting the second-order elastic wave equation
Splitting into longitudinal wave equation and transverse wave equation.
10. The anisotropic media forward system based on stiffness matrix factorization according to claim 8 or 9,
the longitudinal wave equation is:
the transverse wave equation is:
wherein C is p C is a matrix of stiffness coefficients related to longitudinal waves s As a matrix of elastic coefficients related to transverse waves, u p As a longitudinal wave field, u s As a transverse wave field, a full wave field u=u p +u s ρ represents density and t represents time, where G is a gradient matrix in the form of:
wherein x, y, z are space coordinates;
the first order speed-stress equation is:
wherein V is P Representing longitudinal wave velocity vector, V S Represents a transverse wave velocity vector, t represents time, v=v P +V S Representing the total velocity vector, ρ representing density, G representing the gradient matrix, and T representing the stress vector.
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