A forward modeling method and system for anisotropic media based on stiffness matrix decomposition

Abstract

The present invention discloses a forward modeling method and system for anisotropic media based on stiffness matrix decomposition, the method comprising: splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation; deriving a first-order velocity-stress equation for longitudinal wave separation of anisotropic media according to the longitudinal wave equation and the transverse wave equation; and performing forward modeling of anisotropic media according to the first-order velocity-stress equation. The present invention splits the elastic wave equation into a longitudinal wave equation and a transverse wave equation, and then derives a first-order velocity-stress equation for longitudinal and transverse wave separation for anisotropic media according to the split longitudinal wave equation and the transverse wave equation. The amplitude and phase information of the longitudinal and transverse wave fields obtained using the above method are more accurate, and the wave field information is also more complete.

Description

Technical Field

[0001] The invention belongs to the field of petroleum geological exploration, and in particular relates to an anisotropic medium forward modeling method and system based on stiffness matrix decomposition. Background Art

[0002] Anisotropy is widely present in the earth medium. Anisotropic media have the characteristics of P-wave coupling, which makes the forward modeling method based on the isotropic assumption insufficient to accurately describe the propagation law of seismic waves in the actual earth medium. Although the forward modeling of anisotropic elastic wave equation has made certain progress, it also has the following defects: first, the anisotropic parameters are complex, the physical meaning is unclear, and it is difficult to obtain in actual production; second, due to the high cost of multi-component seismic exploration, the current field acquisition is still dominated by P-wave seismic data, and the multi-component elastic wave theory is limited by insufficient data. Therefore, it is necessary to separate the P-wave and S-wave of anisotropic media to meet the needs of seismic data acquisition and processing at the current stage. At present, there are two main methods to extract the longitudinal wave component in anisotropic media. One is the longitudinal and transverse wave field separation method based on the difference in the polarization directions of the longitudinal and transverse waves, which usually adopts the Helmholtz decomposition method, 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 the other is to derive the approximate equation of the pseudo-acoustic wave (qP wave) through the dispersion relationship.

[0003] The propagation direction of seismic waves in anisotropic media is different from that in isotropic media. The longitudinal wave vibration direction is no longer the same as the propagation direction, and the transverse wave vibration direction is perpendicular to the wave field propagation direction. Therefore, the Helmholtz decomposition method based on the difference in the polarization directions of longitudinal and transverse waves cannot achieve the separation of longitudinal and transverse waves well. In addition, in the separation process, the divergence field is used to represent the longitudinal wave and the curl field to represent the transverse wave, so that the vector longitudinal wave field becomes a scalar wave field, which has a different physical meaning from the original wave field, and the amplitude and phase information have changed.

[0004] In terms of the qP wave approximation equation, Alkhalifah (Alkhalifah T. Acoustic approximations for processing in transversely isotropic media. Geophysics, 1998, 63(2): 623-631; Alkhalifah T. An acoustic wave equation for anisotropic media. Geophysics, 2000, 65(4): 1239-1250.) derived the fourth-order pseudo-acoustic wave equation for VTI media based on the precise qP-qSV wave dispersion relationship of VTI media and assumed that the vertical shear wave velocity Vs0 = 0. This is to reduce the complexity of the calculation of the fourth-order partial differential equation. Zhou (Zhou H, Zhang G, Zhang Y. An anisotropic acoustic wave equation for VTI media. 68th EAGE Conference and Exhibition. Vienna: European Association of Geoscientists and Engineers, 2006: 1-5.) and Du (Du X, Fletcher R, Fowler P JA new pseudo-acoustic wave equation for VTI media. 70th EAGE Conference and Exhibition. Rome: European Association of Geoscientists and Engineers, 2008: 1-5.) derived two types of second-order coupled pseudo-acoustic wave equations for VTI media respectively by introducing different auxiliary wave field functions based on the dispersion relationship of acoustic approximation. In order to combine the staggered grid technology to achieve accurate finite difference simulation, Hestholm (Hestholm S. Acoustic VTI modeling using higher-order finite differences. Geophysics, 2009, 74 (5): T67-T73.) derived the pseudo-acoustic wave equation for VTI media in the first-order partial differential form.Duveneck et al. (Duveneck E, Milcik P, Bakker PM. Acoustic VTI wave equations and their application for anisotropic reverse-time migration. SEG Technical Program Expanded Abstracts. Las Vegas: Society of Exploration Geophysicists, 2008: 2186-2190.) also derived the VTI medium pseudo-acoustic wave equations in second-order form and first-order velocity-stress form based on the acoustic approximation of Hooke's law. Similar to Alkhalifah's method, several pseudo-acoustic wave equations all have the problem of low-speed and low-amplitude qSV artificial interference waves (Grechka V., Zhang L, and JWRector, 2004, Shear-waves in acoustic anisotropic media, Geophysics, 69(2): 576-582.; JinS and A.Stovas, 2018, S-wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis, Geophysical Prospecting, 66(6): 1123-1137.), and this interference wave affects the final results of forward simulation and migration imaging. In order to overcome the problem of low-speed and low-amplitude qSV artificial interference waves in the traditional qP wave approximation method, Xu et al. (Xu SB, 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.) proposed a new qP wave approximation method. This method does not assume that the vertical SV wave velocity is zero, but sets the SV wave phase velocity to zero. Based on this condition, the vertical SV wave velocity expression is obtained, and the phase velocity of the qP wave is calculated on this basis.However, the difference between the qP wave phase velocity calculated by this method and the phase velocity calculated by the method of Alkhalifah (Alkhalifah T. An acoustic wave equation for anisotropic media. Geophysics, 2000, 65 (4): 1239-1250.) is very small (eg. Figure 1 (shown), and if the vertical SV wave velocity expression is substituted into the SV wave phase velocity expression, the obtained SV wave phase velocity is not zero. The method proposed by Xu et al. (Xu SB, 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 gives the qP wave phase velocity expression, and does not give the wave field forward modeling results. Another disadvantage of the qP wave approximation method is that the longitudinal wave data of the anisotropic medium is not complete because the information of the converted wave cannot be taken into account.

[0005] The variation of qP wave phase velocity with polarization angle calculated by Xu et al. (2020) and Alkhalifah (2000) is shown in Figure 1 The model parameters are the same as those of Model 1 in Xu et al. (2020) (Vp0 = 3 km / s, Vs0 = 1 km / s, Vpn = 3.3 km / s, anisotropy parameter η = 0.1). Summary of the invention

[0006] In view of the above problems, the present invention proposes an anisotropic medium forward modeling method based on stiffness matrix decomposition, and the forward modeling method comprises the following steps:

[0007] Split the elastic wave equation into the longitudinal wave equation and the transverse wave equation;

[0008] A first-order velocity-stress equation for longitudinal wave separation in anisotropic media is derived based on the longitudinal wave equation and the transverse wave equation;

[0009] Anisotropic medium forward modeling is performed according to the first-order velocity-stress equation.

[0010] Furthermore, splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation comprises the following steps:

[0011] The coefficients C in the stiffness coefficient matrix C 11 , C 22 , C33 , C 12 , C 13 , C 23 , C 44 , C 55 , C 66 Decomposed into a linear function of longitudinal wave modulus and / or shear wave modulus; wherein the stiffness matrix C is in the form of:

[0012]

[0013] It should be noted that the C 11 =C 22 , C 13 =C 23 and C 44 =C 55 , equal coefficients are represented by only one of the coefficients in the stiffness matrix C.

[0014] According to the linear function, the stiffness coefficient matrix C related to the longitudinal wave is established respectively. p The stiffness coefficient matrix C related to shear waves s ;

[0015] According to the C p and C s The second-order elastic wave equation Split into longitudinal wave equation and transverse wave equation.

[0016] Furthermore, 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 is as follows:

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] Where ε is a parameter that measures the anisotropy strength of qS waves; δ is the longitudinal wave velocity V along the symmetry axis.p0 and the longitudinal wave velocity V in the direction perpendicular to the symmetry axis p90 A transitional parameter between γ and γ; γ is a parameter that measures the anisotropy intensity of qS waves or the splitting intensity of shear waves; is the longitudinal wave transverse quantity, is the shear wave quantity, and ρ represents the density.

[0024] Furthermore, the longitudinal wave equation is:

[0025]

[0026] The shear wave equation is:

[0027]

[0028] In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the elastic coefficient matrix related to shear waves, u p is the longitudinal wave field, u s is the shear wave field, and the total wave field u=u p +u s , ρ represents density, t represents time, and G is the gradient matrix in the form of:

[0029]

[0030] Among them, x, y, z are spatial coordinates.

[0031] Further,

[0032] Said

[0033]

[0034] In the formula, C p is the stiffness coefficient matrix related to longitudinal waves; C s is the elastic coefficient matrix related to shear waves; ε is the parameter measuring the anisotropic strength of qS waves; δ is the longitudinal wave velocity V along the symmetry axis p0 and the longitudinal wave velocity V in the direction perpendicular to the symmetry axis p90 A transitional parameter between γ and γ; γ is a parameter that measures the anisotropy intensity of qS waves or the splitting intensity of shear waves; is the longitudinal wave transverse quantity, is the shear wave quantity, and ρ represents the density.

[0035] Furthermore, the first-order velocity-stress equation is:

[0036]

[0037] Where V Prepresents the longitudinal wave velocity vector, V S represents the shear wave velocity vector, t represents time, V = V P +V S represents the total velocity vector, C p is the stiffness coefficient matrix related to longitudinal waves; C s is the elastic coefficient matrix related to shear waves; ρ represents density density, G represents gradient matrix, C represents stiffness coefficient matrix, and T represents stress vector;

[0038] Τ=(σ xx σ yy σ zz σ yz σ xz σ xy ) T

[0039] Among them, σ xx , σ yy and σ zz They represent the normal stress with the principal axis in the x, y, and z directions, respectively, and σ yz , σ xz and σ xy They represent the shear stresses on the yz, xz and xy planes respectively, and the superscript T indicates vector transpose.

[0040] Furthermore, the first-order velocity-stress equation is derived as follows:

[0041] Calculation based on Hooke's law:

[0042] T=CE=CG T u (14)

[0043] Among them, T is the stress vector, C is the stiffness coefficient matrix, E is the strain vector, and u represents the full wave field;

[0044] E=(e xx e yy e zz e yz e xz e xy ) T

[0045] Among them, e xx 、e yy 、e zz represents the normal strain in the x, y, and z directions, e yz 、e xz 、e xy represents the shear strain on the yz, xz, and xy planes, and the superscript T indicates the vector transpose;

[0046] From formula (14), we can see that

[0047] G T u=C -1 Τ (15)

[0048] Let the velocity vector Among them, u p represents the longitudinal wave displacement vector, u s represents the shear wave displacement vector, according to the formula G T u=C -1 T converts the longitudinal wave equation and the transverse wave equation into a first-order velocity-stress equation.

[0049] The present invention also proposes an anisotropic medium forward modeling system based on stiffness matrix decomposition, and the forward modeling system includes the following modules:

[0050] A splitting module is used to split the elastic wave equation into a longitudinal wave equation and a transverse wave equation;

[0051] A derivation module, used for deriving a first-order velocity-stress equation for longitudinal wave separation of anisotropic media according to the longitudinal wave equation and the transverse wave equation;

[0052] The forward modeling module is used to perform forward modeling of anisotropic media according to the first-order velocity-stress equation.

[0053] Furthermore, the splitting module includes:

[0054] Decomposition unit is used to convert 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 decomposed into linear functions of longitudinal wave modulus and / or shear wave modulus;

[0055]

[0056] Establishing a matrix unit for establishing a stiffness coefficient matrix related to longitudinal waves and a stiffness coefficient matrix related to transverse waves respectively according to the linear function;

[0057] Split element to transform the second-order elastic wave equation Split into longitudinal wave equation and transverse wave equation.

[0058] Furthermore, the longitudinal wave equation is:

[0059]

[0060] The shear wave equation is:

[0061]

[0062] In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the elastic coefficient matrix related to shear waves, u p is the longitudinal wave field, u s is the shear wave field, and the total wave field u=u p +u s , ρ represents density, t represents time, where G is the gradient matrix, in the form of:

[0063]

[0064] Among them, x, y, z are spatial coordinates;

[0065] The first-order velocity-stress equation is:

[0066]

[0067] Where V P represents the longitudinal wave velocity vector, V S represents the shear wave velocity vector, t represents time, V = V P +V S represents the total velocity vector, ρ represents the density, G represents the gradient matrix, and T is the stress vector.

[0068] Beneficial effects of the present invention:

[0069] The present invention splits the elastic wave equation into a longitudinal wave equation and a transverse wave equation, and then derives a first-order velocity-stress equation for longitudinal and transverse wave separation for anisotropic media based on the split longitudinal wave equation and transverse wave equation. The amplitude and phase information of the longitudinal and transverse wave fields obtained using the above method are more accurate, and the wave field information is also more complete, which can better perform forward modeling of anisotropic media.

[0070] Other features and advantages of the present invention will be described in the following description, and partly become apparent from the description, or understood by practicing the present invention. The purpose and other advantages of the present invention can be realized and obtained by the structures pointed out in the description, claims and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0072] Figure 1 A schematic diagram showing the variation of the qP wave phase velocity with the polarization angle calculated according to the methods of Xu et al. (2020) and Alkhalifah (2000) in the background technology;

[0073] Figure 2 The forward model used in the embodiment of the present invention is shown;

[0074] Figure 3 The elastic wave field diagram obtained by using a conventional anisotropic elastic wave forward modeling method in an embodiment of the present invention is shown;

[0075] Figure 4 The Y-component and X-component diagrams of the longitudinal wave field and the shear wave field obtained by the Helmholtz decomposition method in an embodiment of the present invention are shown;

[0076] Figure 5 The qP wave forward modeling result obtained by using the pseudo-acoustic wave approximation method in an embodiment of the present invention is shown;

[0077] Figure 6 The longitudinal wave field diagram obtained by using the anisotropic medium forward modeling method based on stiffness coefficient matrix decomposition proposed by the present invention in the embodiment of the present invention is shown;

[0078] Figure 7 The shear wave field diagram obtained by using the anisotropic medium forward modeling method based on stiffness coefficient matrix decomposition proposed in the embodiment of the present invention is shown. DETAILED DESCRIPTION

[0079] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0080] In order to obtain the longitudinal wave data of anisotropic media with relatively accurate amplitude and phase information and relatively complete wave field information, the present invention derives the first-order velocity-stress equation for the separation of longitudinal and transverse waves in anisotropic media by approximating the coupling term in the stiffness coefficient matrix as a linear function of the longitudinal wave modulus and the transverse wave modulus, thereby realizing the separation of longitudinal and transverse wave fields in anisotropic media. Compared with other wave field separation algorithms, the amplitude and phase information of the longitudinal and transverse wave fields obtained by this "equation separation" algorithm are more accurate, and the wave field information is also more complete.

[0081] The present invention proposes an anisotropic medium forward modeling method based on stiffness matrix decomposition, and the forward modeling method comprises the following steps:

[0082] Step 1: Split the elastic wave equation into the longitudinal wave equation and the transverse wave equation;

[0083] The wave equation is represented by the stiffness matrix C. After decomposing the stiffness matrix C into longitudinal wave components and transverse wave components, the wave equation can be decomposed into longitudinal wave equation and transverse wave equation. The stiffness coefficient matrix C is decomposed into a linear function of the longitudinal wave modulus and the transverse wave modulus; specifically, the stiffness coefficient matrix C can be decomposed into a part related to longitudinal waves and a part related to transverse waves, thereby splitting the elastic wave equation into two parts: the longitudinal wave equation and the transverse wave equation.

[0084] Among them, the stiffness coefficient matrix is ​​in the form of:

[0085]

[0086] In the formula, C 11 , C 22 , C 33 , C 12 , C 13 , C 23 , C 44 , C 55 , C 66 Represents the coefficients of the stiffness matrix.

[0087] It should be noted that the C 11 =C 22 , C 13 =C 23 and C 44 =C 55 , equal coefficients are represented by only one of the coefficients in the stiffness matrix C.

[0088] In the embodiment of the present invention, the splitting process is described by taking VTI media as an example.

[0089] For VTI media, the components with non-zero stiffness coefficients are expressed by Thomsen parameters as follows:

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096] Among them, C 11 , C 22 , C 33 , C 12 , C 13 , C 23 , C 44 , C 55 , C 66 are the coefficients in the stiffness matrix C in equation (1).

[0097] Formulas (2)-(7) are written in matrix form, as shown in formula (8):

[0098]

[0099] Where V p0 and V s0 are the phase velocities of qP waves and qSV waves along the symmetry axis of the VTI medium, respectively. ε, δ and γ are three dimensionless factors that represent the anisotropy strength of the VTI medium. Among them, ε is a parameter that measures the anisotropy strength of qS waves. The larger the ε, the greater the anisotropy strength of the longitudinal waves of the medium. δ is the longitudinal wave velocity V along the symmetry axis. p0 and the longitudinal wave velocity V in the direction perpendicular to the symmetry axis p90 A transitional parameter between γ and ρ; γ can be regarded as a parameter to measure the anisotropy intensity of qS waves or the shear wave splitting intensity. The larger γ is, the greater the shear wave anisotropy of the medium is. ρ represents the density.

[0100] From equations (2)-(7), it can be seen that except for C 13 and C 23 In addition, the coefficients are all longitudinal wave moduli and shear wave modulus A linear function of 13 and C 23 Expressed as the longitudinal wave modulus and shear wave modulus The stiffness coefficient matrix C can be decomposed into the part C related to the longitudinal wave. p and the part C related to the shear wave s .

[0101] In step 1, splitting the elastic wave equation into the longitudinal wave equation and the transverse wave equation includes the following steps:

[0102] 1. Decompose the stiffness coefficient matrix C into a linear function of the longitudinal wave modulus and the transverse wave modulus;

[0103] Since the stiffness matrix coefficient C 13 and C 23 Not the longitudinal wave modulus and / or shear wave modulus is a linear function, so we can first 13 and C 23 The expression (5) is transformed to obtain the expression (9)

[0104]

[0105] In formula (9),

[0106]

[0107] In most parameter configurations, |Δ| < 1, because The maximum value of is 0.707, so The maximum value of is about 0.25, which is similar to Compared to the small amount, so can be This term is discarded from the formula Δ, and Δ can be transformed into:

[0108]

[0109] Taylor expansion of equation (11) yields

[0110]

[0111] Therefore, the stiffness coefficient C 13 and C 23 Converted to longitudinal wave modulus and shear wave modulus A linear function of .

[0112] 2. According to the linear function, the stiffness coefficient matrix C related to the longitudinal wave is established respectively p The stiffness coefficient matrix C related to shear waves s .

[0113] The stiffness coefficient matrix C associated with longitudinal waves in VTI media p The stiffness coefficient matrix C related to shear waves s They are:

[0114]

[0115]

[0116] 3. Thus, the second-order elastic wave equation can be It is split into the longitudinal wave equation and the transverse wave equation in equations (15)-(16).

[0117] The split longitudinal wave equation and transverse wave equation are as follows:

[0118] The longitudinal wave equation is:

[0119] The shear wave equation is:

[0120] In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the elastic coefficient matrix related to shear waves, u p and u s are longitudinal wave field and transverse wave field respectively, and the total wave field u=u p +u s , ρ represents density, t represents time, superscript T represents vector transpose, and G is the gradient matrix, which is in the form of:

[0121]

[0122] Among them, x, y, z are spatial coordinates.

[0123] Step 2: Based on the split longitudinal wave equation and shear wave equation, derive the first-order velocity-stress equation for longitudinal and shear wave separation in anisotropic media.

[0124] In order to facilitate the application of the staggered grid method for seismic forward modeling, the P-wave equation and S-wave equation in equations (15) and (16) are transformed into the form of first-order velocity-stress equations.

[0125] The first-order velocity-stress equation is obtained by following the steps below:

[0126] Calculate according to Hooke's law. The expression of Hooke's law is

[0127] T=CE=CG T u (18)

[0128] Where, T=[σ 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 the strain vector; where σ xx , σ yy , σ zz They represent the normal stress with the principal axis in the x, y, and z directions, respectively, and σ yz , σ xz and σ xydenote the shear stress on the yz, xz, and xy planes respectively, and the superscript T denotes the vector transpose, where e xx 、e yy 、e zz represents the normal strain in the x, y, and z directions, e yz 、e xz 、e xy represents the shear strain on the yz, xz, and xy planes, the superscript T represents the vector transpose, and u represents the full wave field;

[0129] From formula (18), we can see that

[0130] G T u=C -1 Τ (19)

[0131] Let the velocity vector Using formula (19), u p and u s are longitudinal wave field and transverse wave field respectively, and the total wave field u=u p +u s Equations (15) and (16) can be transformed into the first-order velocity-stress equations in the following form:

[0132]

[0133] Where V P represents the longitudinal wave velocity vector, V S represents the shear wave velocity vector, t represents time, V = V P +V S represents the total velocity vector, ρ represents the density, G represents the gradient matrix, and T is the stress vector.

[0134] Example

[0135] The anisotropic medium forward modeling method based on stiffness matrix decomposition proposed in the present invention is tested. The above method is tested using a three-layer horizontal layered medium model. The model parameters include five stiffness coefficients C 11 , C 33 , C 13 , C 44 , C 66 A three-dimensional model is used, the model size is 2400m(x)×2370m(y)×2700m(z), and the spatial grid size in the x, y, and z directions is 15m. Figure 2 shown.

[0136] The simulation conditions are as follows: the simulated physical time is 1s, the time step is 0.5ms, a Ricker wavelet source is used, the main frequency is 30Hz, and the source is located at x=1350m, y=1185m, z=465m.

[0137] The above simulation conditions were used to Figure 2 The conventional anisotropic elastic wave forward modeling is performed on the model, and the wave field obtained is as follows Figure 3 As shown, Figure 3 From left to right, they are Z component, Y component and X component. Figure 3 The wave fields obtained on each component contain both longitudinal and shear wave data. Since the impedance difference between the third layer and the second layer is very small, the corresponding reflection signal is very weak and almost invisible in the figure.

[0138] Using the Helmholtz decomposition method Figure 2 The mixed wave field in the image is separated into longitudinal and transverse waves, and the results are as follows Figure 4 As shown in Figure 2, the longitudinal wave field obtained by the Helmholtz decomposition method is a scalar field with only one component. Figure 4 The Y and X components of the separated shear wave field are shown in Figure 2. Figure 4 and Figure 3 It can be seen that the longitudinal wave and shear wave data obtained by the Helmholtz decomposition method are distorted in both amplitude and phase compared with the original wave field.

[0139] Using the pseudo-acoustic wave approximation forward modeling method of Alkhalifah (2000), the three components of the pseudo-acoustic wave are obtained as follows: Figure 5 As shown in the figure, from left to right are the Z component, Y component and X component. Figure 5 It can be seen that there are SV interference waves in the longitudinal wave data obtained by the forward modeling method based on the pseudo-acoustic wave approximation (shown as S1 in the figure). Figure 3 Compared with the longitudinal wave field in the original wave field, it can be seen that the longitudinal wave data obtained by the pseudo-acoustic approximation method are different from the real longitudinal waves in amplitude and phase, and the pseudo-acoustic approximation method cannot obtain the converted longitudinal wave information (P2) in the original wave field.

[0140] The anisotropic medium forward modeling method based on stiffness coefficient matrix decomposition proposed in this invention is applied to Figure 2 The parameter model in the figure can be used to obtain the longitudinal wave field data and the shear wave field data. Figure 6 and Figure 7 As shown, Figure 6 and Figure 7 From left to right in the figure are Z component, Y component and X component respectively. Comparing the forward modeling results of the present invention with those of the Helmholtz decomposition method and the pseudo-acoustic wave approximation method, it can be seen that the anisotropic medium forward modeling method based on the stiffness coefficient matrix decomposition can obtain longitudinal wave data and transverse wave data with accurate amplitude and phase information.

[0141] From the above comparison results, it can be seen that the anisotropic medium forward modeling method based on stiffness coefficient matrix decomposition proposed in this invention is applied to Figure 2The parameter model in can obtain more accurate amplitude and phase information of the longitudinal and transverse wave fields as well as the wave field information.

[0142] Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent substitutions for some of the technical features therein; and these modifications or substitutions do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A forward modeling method for anisotropic media based on stiffness matrix decomposition. It is characterized in that The forward modeling method comprises the following steps: Split the elastic wave equation into the longitudinal wave equation and the transverse wave equation; A first-order velocity-stress equation for separation of longitudinal and transverse waves in anisotropic media is derived based on the longitudinal wave equation and the transverse wave equation; Performing forward modeling of anisotropic media according to the first-order velocity-stress equation; The step of splitting the elastic wave equation into a longitudinal wave equation and a transverse wave equation comprises the following steps: The coefficients C in the stiffness coefficient matrix C 11 , C 22 , C 33 , C 12 , C 13 , C 23 , C 44 , C 55 , C 66 Decomposed into a linear function of longitudinal wave modulus and / or shear wave modulus; wherein the stiffness coefficient matrix C is in the form of: According to the linear function, the stiffness coefficient matrix C related to the longitudinal wave is established respectively. p and the stiffness coefficient matrix C associated with shear waves s ; According to C p and C s The second-order elastic wave equation Split into longitudinal wave equation and transverse wave equation; In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the stiffness coefficient matrix related to shear waves, u p is the longitudinal wave field, u s is the shear wave field, and the total wave field u=u p +u s , ρ represents density, t represents time, G is the gradient matrix, and the form is: Among them, x, y, z are spatial coordinates.

2. The anisotropic medium forward modeling method based on stiffness matrix decomposition according to claim 1, It is characterized in that 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 is as follows: Where ε is a parameter that measures the anisotropy strength of qP waves; δ is the longitudinal wave velocity V along the symmetry axis. p0 and the longitudinal wave velocity V in the direction perpendicular to the symmetry axis p90 A transitional parameter between γ and γ; γ is a parameter that measures the anisotropy intensity of qS waves or the splitting intensity of shear waves; is the longitudinal wave modulus, is the shear wave modulus, and ρ represents the density.

3. The anisotropic medium forward modeling method based on stiffness matrix decomposition according to claim 1, It is characterized in that The longitudinal wave equation is: The shear wave equation is: In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the stiffness coefficient matrix related to shear waves, u p is the longitudinal wave field, u s is the shear wave field, and the total wave field u=u p +u s , ρ represents density, t represents time, G is the gradient matrix, and the form is: Among them, x, y, z are spatial coordinates.

4. The anisotropic medium forward modeling method based on stiffness matrix decomposition according to claim 3, It is characterized in that In the formula, C p is the stiffness coefficient matrix related to longitudinal waves; C s is the stiffness coefficient matrix related to shear waves; ε is a parameter that measures the anisotropy strength of qP waves; δ is the longitudinal wave velocity V along the symmetry axis. p0 and the longitudinal wave velocity V in the direction perpendicular to the symmetry axis p90 A transitional parameter between γ and γ; γ is a parameter that measures the anisotropy intensity of qS waves or the splitting intensity of shear waves; is the longitudinal wave modulus, is the shear wave modulus, and ρ represents the density.

5. The anisotropic medium forward modeling method based on stiffness matrix decomposition according to claim 1, It is characterized in that The first-order velocity-stress equation is: Where V P represents the longitudinal wave velocity vector, V S represents the shear wave velocity vector, t represents time, V = V P +V S represents the total velocity vector, C p is the stiffness coefficient matrix related to longitudinal waves; C s is the stiffness coefficient matrix related to shear waves; ρ represents density, G represents gradient matrix, C represents stiffness coefficient matrix, and M represents stress vector; M=(σ xx s yy s zz s yz s xz s xy ) T Among them, σ xx , σ yy and σ zz They represent the normal stress with the principal axis in the x, y, and z directions, respectively, and σ yz , σ xz and σ xy They represent the shear stresses on the yz, xz and xy planes respectively, and the superscript T indicates vector transpose.

6. The anisotropic medium forward modeling method based on stiffness matrix decomposition according to claim 5, It is characterized in that The first-order velocity-stress equation is derived as follows: Calculation based on Hooke's law: M=CE=CG T u (14) Among them, M is the stress vector, C is the stiffness coefficient matrix, E is the strain vector, and u represents the full wave field; And=(and xx And yy And zz And yz And xz And xy ) T Among them, e xx 、e yy 、e zz represents the normal strain in the x, y, and z directions, e yz 、e xz 、e xy represents the shear strain on the yz, xz, and xy planes, and the superscript T indicates the vector transpose; From formula (14), we can see that G T u=C -1 M (15) Let the velocity vector Among them, u p represents the longitudinal wave field, u s represents the shear wave field, according to the formula G T u=C -1 M transforms the longitudinal wave equation and the transverse wave equation into a first-order velocity-stress equation.

7. A forward modeling system for anisotropic media based on stiffness matrix decomposition. It is characterized in that The forward modeling system includes the following modules: A splitting module is used to split the elastic wave equation into a longitudinal wave equation and a transverse wave equation; A derivation module, used for deriving a first-order velocity-stress equation for longitudinal and transverse wave separation of anisotropic media according to the longitudinal wave equation and the transverse wave equation; A forward modeling module, used for performing anisotropic medium forward modeling according to the first-order velocity-stress equation; The splitting module comprises: Decomposition unit is used to convert 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 Decomposed into a linear function of longitudinal wave modulus and / or shear wave modulus; wherein the stiffness coefficient matrix C is in the form of: Establishing a matrix unit for establishing a stiffness coefficient matrix associated with longitudinal waves and a stiffness coefficient matrix associated with transverse waves according to the linear function; Split element to transform the second-order elastic wave equation Split into longitudinal wave equation and transverse wave equation; According to C p and C s The second-order elastic wave equation Split into longitudinal wave equation and transverse wave equation; In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the stiffness coefficient matrix related to shear waves, u p is the longitudinal wave field, u s is the shear wave field, and the total wave field u=u p +u s , ρ represents density, t represents time, G is the gradient matrix, and the form is: Among them, x, y, z are spatial coordinates.

8. The anisotropic medium forward modeling system based on stiffness matrix decomposition according to claim 7, It is characterized in that The longitudinal wave equation is: The shear wave equation is: In the formula, C p is the stiffness coefficient matrix related to longitudinal waves, C s is the stiffness coefficient matrix related to shear waves, u p is the longitudinal wave field, u s is the shear wave field, and the total wave field u=u p +u s , ρ represents density, t represents time, where G is the gradient matrix, in the form of: Among them, x, y, z are spatial coordinates; The first-order velocity-stress equation is: Where V P represents the longitudinal wave velocity vector, V S represents the shear wave velocity vector, t represents time, V = V P +V S represents the total velocity vector, ρ represents the density, G represents the gradient matrix, M represents the stress vector, and the superscript T represents the vector transpose.

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