Pseudo Helmholtz decomposition method, device and medium for longitudinal and transverse waves in anisotropic media

Abstract

The present invention provides a method, apparatus and medium for pseudo-Helmholtz decomposition of longitudinal and transverse waves in an anisotropic medium. The method includes: S1. Obtaining a first-order pseudo-Helmholtz decomposition formula, a decomposition operator and its Poisson equation; S2. Transforming the first-order pseudo-Helmholtz decomposition formula, the decomposition operator and the Poisson equation into the wavenumber domain and constructing a new pseudo-Helmholtz decomposition formula, a decomposition operator and a Poisson equation using the wavefront phase direction; S3. Transforming the new pseudo-Helmholtz decomposition formula, the decomposition operator and the auxiliary equation obtained in the wavenumber domain back to the spatial domain, and the P / S wave decomposition formula of the spatial domain anisotropic medium can be obtained. The method uses the wavefront phase direction vector to obtain a new pseudo-Helmholtz decomposition formula, a decomposition operator and the Poisson equation satisfied by the auxiliary wave field, and has better effects than the zero-order pseudo-Helmholtz decomposition operator and less computational complexity.

Description

Technical Field

[0001] The present invention provides a method, an apparatus, a medium and a device for pseudo Helmholtz decomposition of longitudinal and transverse waves in an anisotropic medium, belonging to the field of elastic wave imaging. Background Art

[0002] Currently, reverse time migration (RTM) is the best imaging method for complex structures and is widely used in actual production, which can improve imaging quality and generate great value. However, these applications mainly focus on the imaging of PP waves (P-wave incidence, P-wave reflection). In reality, most rocks on the earth are anisotropic elastic media. Therefore, the observed seismic data includes PP waves (P-wave incidence, P-wave reflection), PS waves (P-wave incidence, S-wave reflection), SS waves (S-wave incidence, S-wave reflection), and SP waves (S-wave incidence, P-wave reflection). Here, the P-wave refers to the longitudinal wave, and the S-wave refers to the transverse wave. Moreover, the S-wave is divided into fast and slow S-waves. Since the S-wave is insensitive to fluids, it is less affected by fluids such as oil and natural gas. Therefore, elastic wave imaging considering the S-wave is of great significance. Since the P-wave and the S-wave are coupled and multi-component during the propagation of elastic waves, if the elastic wave equation is not decoupled, there will be crosstalk during migration imaging, resulting in artifacts and affecting imaging quality. Therefore, to achieve elastic wave imaging, it is necessary to first decouple and separate the P-wave and the S-wave, and then image the decoupled wave fields respectively to obtain PP migration imaging, PS migration imaging, SS migration imaging, and SP migration imaging.

[0003] When elastic waves propagate in an isotropic medium, the propagation direction of the wave is consistent with the polarization direction of the P-wave and perpendicular to the polarization direction of the S-wave. However, when elastic waves propagate in an anisotropic medium, the propagation direction of the wave is neither parallel nor perpendicular to the polarization direction of the P / S-wave. In an anisotropic medium, a main idea for separating P and S waves is to project the vibration vector field of the particles onto the polarization directions of the P and S waves. If the projected energy is consistent with the polarization direction of this type of wave, then this type of wave is separated. Dellinger and Etgen (in 1990) obtained the polarization directions of P / S waves by solving the Christoffel equation, and then projected the wave field onto the polarization directions of P / S waves respectively to achieve the separation of P / S waves in an anisotropic medium. Zhang and McMechan (in 2010) realized the wave field separation of VTI media in the wavenumber domain based on the Christoffel equation and the Helmholtz method. Yang (in 2019) proposed a pseudo-derivative operator for separating P / S waves in an anisotropic medium in the spatial domain. In this article, the first-order pseudo-Helmholtz decomposition operator and the elliptical anisotropy (zero-order) pseudo-Helmholtz decomposition operator were mentioned to separate P / S waves. However, the separation effect of the zero-order pseudo-Helmholtz decomposition operator is not ideal when the anisotropy parameters differ greatly, and the first-order pseudo-Helmholtz decomposition operator requires multiple solutions of the Poisson equation in the spatial domain, which increases the computational amount. Summary of the Invention

[0004] Aiming at the above problems, the purpose of the present invention is to provide a pseudo-Helmholtz decomposition method, device, medium and equipment for longitudinal and transverse waves in an anisotropic medium, so as to solve the problems that the existing zero-order pseudo-Helmholtz decomposition operator decomposes the anisotropic wave field, and when the anisotropy parameter difference is large, the decomposition effect is poor and there are many residues, while using the first-order pseudo-Helmholtz decomposition operator requires multiple solutions of the Poisson equation in the spatial domain, which increases the computational amount.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions:

[0006] In the first aspect, the present invention provides a pseudo-Helmholtz decomposition method for longitudinal and transverse waves in an anisotropic medium, including:

[0007] Obtain the three-dimensional first-order pseudo-Helmholtz decomposition formula, decomposition operator and its Poisson equation;

[0008] Transform the three-dimensional first-order pseudo-Helmholtz decomposition formula, decomposition operator and its Poisson equation into the wavenumber domain and use the wavefront phase direction to construct a new pseudo-Helmholtz decomposition formula, new decomposition operator and its new Poisson equation, including:

[0009] Transform the three-dimensional first-order pseudo-Helmholtz decomposition formula, decomposition operator and Poisson equation into the wavenumber domain;

[0010] Calculate the wave field u=(ux , u y , u z ) The wavefront phase direction vector n of, where n includes n1, n2, n3, and n1, n2, n3 respectively correspond to the wave field u x , u y , u z ; the wavefront phase direction vector of

[0011] Represent a new pseudo - Helmholtz decomposition formula, a new decomposition operator, and a new Poisson equation according to the wavefront phase direction vector n;

[0012] Transform the new pseudo - Helmholtz decomposition formula, the new decomposition operator, and the new auxiliary equation back to the spatial domain to obtain the P - S wave decomposition formula for anisotropic media in the spatial domain, and perform P - S wave separation according to the P - S wave decomposition formula.

[0013] Furthermore, the three - dimensional first - order pseudo - Helmholtz decomposition formula, the decomposition operator, and its Poisson equation are respectively:

[0014]

[0015] Among them, u represents the input wave field, which is a vector wave field mixed with P - wave and S - wave, and has three components for a three - dimensional model, u p , u s = u SH + u SV , respectively represent the vector wave fields of P - wave and S - wave, and the S - wave includes SH - wave and SV - wave; respectively represent taking spatial derivatives along the x, y, z directions; w is an auxiliary wave field obtained from the input wave field u; v p , v s respectively represent the longitudinal and transverse wave velocities of the wave along the symmetry axis direction of the transversely anisotropic medium, ε, δ are dimensionless factors representing the strength of anisotropy, and the expressions of r1(x), r2(x), r3(x), r4(x) are respectively:

[0016]

[0017] Furthermore, transforming the first - order pseudo - Helmholtz decomposition operator to the wavenumber domain, the P - S wave decomposition formula, the decomposition operator, and the Poisson equation in the wavenumber domain are respectively:

[0018]

[0019] Or respectively:

[0020]

[0021] Among them, U, U p , U s, W are u, u p , u s , w are the input wave fields, longitudinal wave vectors, transverse wave vectors, and auxiliary wave fields in the corresponding wavenumber domain obtained by Fourier transform; k x , k z represent the wavenumbers in the x and z directions; When the anisotropic parameters are constants, they are r1, r2, r3, r4 respectively.

[0022] Further, using the wavefront phase direction, the P - S wave decomposition formula, decomposition operator, and Poisson equation in the wavenumber domain are constructed into a new pseudo - Helmholtz decomposition formula, new decomposition operator, and new auxiliary equation, and the P - S wave separation formula in the anisotropic medium in the spatial domain is obtained, including:

[0023] Define the wavefront phase direction vector n = (n x , n y , n z ) And make the following substitutions for the Poisson equation in Equation (2) or Equation (3):

[0024] Obtain:

[0025]

[0026] Calculate the wavefront phase direction vector n of the wave field u = (u x , u y , u z ), the n includes n1, n2, n3, and the n1, n2, n3 respectively correspond to the wavefront phase direction vectors of the wave field u x , u y , u z The calculation formula of n is as Equation (5):

[0027]

[0028] According to the wavefront phase direction vector n, the decomposition formula and decomposition operator in Equation (3) are also deformed as the Poisson equation to Equation (4) to obtain the new pseudo - Helmholtz decomposition formula, new decomposition operator, and new Poisson equation respectively as:

[0029]

[0030] Perform the inverse Fourier transform on Equation (6) to obtain the P - S wave decomposition formula, decomposition operator, and Poisson equation in the spatial domain, that is, the P - S wave decomposition formula in the anisotropic medium in the spatial domain is:

[0031]

[0032] wherein, u represents the input wave field, which is a vector wave field mixed with P-wave and S-wave, and has three components for a three-dimensional model, u p , u s = u SH + u SV , respectively representing the vector wave fields of P-wave and S-wave respectively representing the spatial derivatives along the x, y, and z directions. w is the auxiliary wave field obtained from the input wave field u, v p , v s respectively representing the longitudinal wave and transverse wave velocities of the wave along the symmetry axis direction of the transversely isotropic medium. ε and δ are dimensionless factors representing the strength of anisotropy. The expressions of r1(x), r2(x), r3(x), and r4(x) are the same as those in formula (1). n = (n x , n y , n z ) represents the wavefront phase direction vector of the corresponding component of the input wave field

[0033] Furthermore, it also includes: neglecting the derivative term along the y direction in formula (7), the anisotropic wave field decomposition formula in two-dimensional case can be obtained as follows

[0034]

[0035] wherein, u represents the input wave field, which is a vector wave field mixed with P-wave and S-wave, u p , u s respectively representing the vector wave fields of P-wave and S-wave respectively representing the spatial derivatives along the x and z directions. w is the auxiliary wave field obtained from the input wave field u. v p , v s respectively representing the longitudinal wave and transverse wave velocities of the wave along the symmetry axis direction of the transversely isotropic medium. ε and δ are dimensionless factors representing the strength of anisotropy. The expressions of r1(x), r2(x), r3(x), and r4(x) are the same as those in formula (1). n = (n x , n z ) represents the wavefront phase direction vector of the corresponding component of the input wave field

[0036] In the second aspect, the present invention also provides a pseudo Helmholtz decomposition device for longitudinal and transverse waves in an anisotropic medium, including

[0037] The first processing unit is used to obtain the first-order pseudo Helmholtz decomposition formula, the decomposition operator and its Poisson equation

[0038] The second processing unit is used to transform the first-order pseudo Helmholtz decomposition operator into the wavenumber domain by calculating the wave field u = (u x , u y , u zThe wavefront phase direction vector n of (), where n includes n1, n2, and n3, and n1, n2, and n3 respectively correspond to the wave field u x , u y , u z the wavefront phase direction vector, and use the direction of the wavefront phase vector n to construct a new pseudo Helmholtz decomposition formula, a new decomposition operator, and a new Poisson equation;

[0039] The third processing unit transforms the new pseudo Helmholtz decomposition formula, the new decomposition operator, and the new auxiliary equation obtained in the wavenumber domain back to the spatial domain, and then the P - S wave decomposition formula for anisotropic media in the spatial domain can be obtained;

[0040] In a third aspect, the present invention also provides a computer - readable storage medium storing computer instructions, and the computer instructions are used to implement the pseudo Helmholtz decomposition method of P - S waves in anisotropic media when being executed by a processor.

[0041] In a fourth aspect, the present invention also provides a computer device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. It is characterized in that when the processor executes the computer program, the pseudo Helmholtz decomposition method of P - S waves in anisotropic media is implemented.

[0042] Due to the above - mentioned technical solutions adopted by the present invention, it has the following advantages:

[0043] Using the wavefront phase direction vector to calculate the first - order pseudo Helmholtz decomposition operator and the Poisson equation satisfied by the auxiliary wave field, a P - S wave field decomposition method for anisotropic media is obtained, which has better effects than the zero - order pseudo Helmholtz decomposition operator and smaller computational complexity than the first - order pseudo Helmholtz decomposition operator.

[0044] It can achieve high - precision P / S wave field decomposition in elliptical / non - elliptical anisotropic media. Its accuracy and effect are much higher than those of the "zero - order / first - order pseudo Helmholtz decomposition operator", and it does not require as much computational complexity as the "first - order pseudo Helmholtz decomposition operator". It significantly improves both the computational efficiency and the separation effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] By reading the detailed description of the preferred embodiments below, various other advantages and benefits will become clear to those of ordinary skill in the art. The drawings are only for the purpose of showing the preferred embodiments and are not considered to be a limitation of the present invention. Throughout the drawings, the same reference numerals are used to represent the same components.

[0046] In the drawings:

[0047] Figure 1 is a flowchart of the pseudo Helmholtz decomposition method of P - S waves in anisotropic media provided by the present invention;

[0048] Figure 2 are the horizontal and vertical components of the homogeneous model velocity wavefield;

[0049] Figure 3 are the P / S waves obtained by separating the wavefield of Figure 2 using the zero-order pseudo Helmholtz decomposition operator of Yang (2019);

[0050] Figure 4 are the P / S waves obtained by separating the wavefield of Figure 2 using the new pseudo Helmholtz decomposition operator proposed by the present invention;

[0051] Figure 5 are the vertical P / S wave velocities, medium density, and anisotropy parameter models of the partial Hess model;

[0052] Figure 6 are Figure 5 the horizontal and vertical components of the velocity wavefield obtained using the model parameters given in;

[0053] Figure 7 are the P waves obtained by separating the wavefield of Figure 6 using the zero-order pseudo Helmholtz decomposition operator of Yang (2019);

[0054] Figure 8 are the P waves obtained by separating the wavefield of Figure 6 using the new pseudo Helmholtz decomposition operator proposed by the present invention;

[0055] Figure 9 are the S waves obtained by separating the wavefield of Figure 6 using the zero-order pseudo Helmholtz decomposition operator of Yang (2019);

[0056] Figure 10 are the S waves obtained by separating the wavefield of Figure 6 using the new pseudo Helmholtz decomposition operator proposed by the present invention. Detailed implementation manners

[0057] The exemplary embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. Although the exemplary embodiments of the present invention are shown in the drawings, it should be understood that the present invention can be implemented in various forms and should not be limited by the embodiments set forth herein. On the contrary, these embodiments are provided so that the present invention can be more thoroughly understood and the scope of the present invention can be fully conveyed to those skilled in the art.

[0058] The present invention provides a Helmholtz decomposition method for longitudinal and transverse waves in an anisotropic medium. The method uses the wavefront phase direction vector to calculate the new pseudo-Helmholtz decomposition operator and the Poisson equation satisfied by the auxiliary wave field, obtaining a method for decomposing the longitudinal and transverse wave fields in an anisotropic medium that has a better effect than the zero-order pseudo-Helmholtz decomposition operator and a smaller computational amount than the first-order pseudo-Helmholtz decomposition operator.

[0059] As Figure 1 shown, Embodiment 1 of the present invention provides a pseudo-Helmholtz decomposition method for longitudinal and transverse waves in an anisotropic medium, including the steps:

[0060] S1. Obtain the three-dimensional first-order pseudo-Helmholtz decomposition formula, decomposition operator, and its Poisson equation;

[0061] For the two-dimensional case, the first-order pseudo-Helmholtz decomposition formula is prior art, and the first-order pseudo-Helmholtz decomposition operator in the three-dimensional case can be obtained by analogy. The specific calculation process of the two-dimensional first-order pseudo-Helmholtz decomposition is as follows:

[0062] Yang (in 2022) decomposed the P / S waves in VTI media mainly based on the Christoffel equation and the Helmholtz method. First, it is necessary to solve the polarization directions of the P / S waves through the Christoffel equation in the wavenumber domain. Then transform it back to the spatial domain and project the wave vector onto the polarization directions of the P / S waves, thereby realizing the decomposition of the P / S waves in VTI media.

[0063] When the VTI medium is elliptically anisotropic, then ε = δ. At this time, according to the polarization directions of the P / S waves, the formed P / S wave separation operator is defined as the "zero-order pseudo-Helmholtz decomposition operator":

[0064]

[0065] where respectively represent taking the spatial derivatives along the x and z directions. v p , v s respectively represent the longitudinal and transverse wave velocities of the wave along the symmetry axis direction of the transversely anisotropic medium. ε and δ are dimensionless factors representing the strength of anisotropy.

[0066] Then the decomposition formula of the P / S waves is

[0067]

[0068] where the auxiliary wave field w0 satisfies the following Poisson equation

[0069]

[0070] where, here u represents the input wave field, which is a vector wave field mixed with P waves and S waves. For a two-dimensional model, it has two components, up , u s represent the vector wave fields of P-wave and SV-wave respectively.

[0071] When the VTI medium is non-elliptical anisotropic, then ε≠δ. At this time, according to the polarization directions of P / S waves, it is defined as the "first-order pseudo Helmholtz decomposition operator":

[0072]

[0073] where

[0074]

[0075]

[0076] The decomposition formula of P / S wave is

[0077]

[0078] where the auxiliary wave field w satisfies the following Poisson equation

[0079]

[0080] where, here u represents the input wave field, which is a vector wave field mixed with P-wave and S-wave. For a two-dimensional model, it has two components, u p , u s represent the vector wave fields of P-wave and SV-wave respectively.

[0081] It can be seen from formula (6) that it is very difficult to directly solve the Poisson equation of the first-order pseudo Helmholtz decomposition operator in the spatial domain. Yang (in 2019) mentioned that when , formula (6) and (7) are equivalent. And because is much smaller than r1(x), so it is reasonable to make the following approximation to the Poisson equation.

[0082]

[0083] Using formula (5) and (7), the wave field separation of VTI medium in the spatial domain can be realized. First, the auxiliary wave field w is obtained through the formula, and then w is substituted into formula (5) to realize wave field separation. However, when applying the first-order pseudo Helmholtz decomposition operator to w, first ( needs to be calculated. Since the derivative is in the denominator, it is very difficult to solve. Taking as an example, for this term, first calculate the following Poisson equation

[0084]

[0085] Solving formula (8) can obtain w′, and then find the mixed partial derivative to achieve the processing of It can be seen that each time the operator acts, the vector anisotropic Poisson equation needs to be solved once. Although this processing method can obtain a better separation effect compared with the zero-order pseudo-Helmholtz separation operator, the computational cost is very high.

[0086] Therefore, the obtained first-order pseudo-Helmholtz decomposition formula, decomposition operator, and Poisson equation are shown in Equation (9):

[0087]

[0088] where u represents the input wave field, which is a vector wave field mixed with P-wave and S-wave. For a two-dimensional model, it has two components, u p , u s respectively represent the vector wave fields of P-wave and SV-wave. w is the auxiliary wave field required to solve the P / S wave. The expressions of r1(x), r2(x), r3(x), and r4(x) are the same as above.

[0089] Through the two-dimensional first-order pseudo-Helmholtz decomposition method derived by Yang (in 2019) and the three-dimensional zero-order pseudo-Helmholtz decomposition method derived by Zuo (in 2022), the three-dimensional first-order pseudo-Helmholtz decomposition method is derived here

[0090]

[0091] S2. Transform the three-dimensional first-order pseudo-Helmholtz decomposition formula, decomposition operator, and Poisson equation into the wavenumber domain and use the wavefront phase direction to construct a new pseudo-Helmholtz decomposition formula, new decomposition operator, and new Poisson equation, including:

[0092] (1) Transform the three-dimensional first-order pseudo-Helmholtz decomposition formula, decomposition operator, and Poisson equation into the wavenumber domain to obtain:

[0093]

[0094] where u represents the input wave field, which is a vector wave field mixed with P-wave and S-wave. For a three-dimensional model, it has three components, u p , u s = u SH + u SV , respectively represent the vector wave fields of P-wave and S-wave (including SH-wave and SV-wave). w is the auxiliary wave field required to solve the P / S wave. The expressions of r1(x), r2(x), r3(x), and r4(x) are the same as those in Equation (1).

[0095] Equation (11) can also be expressed in the following form

[0096]

[0097] Its parameter representation meaning is the same as above.

[0098] (2) Define the wavefront phase direction vector n = (n x , n y , n z ) And make the following substitutions for the formula (12):

[0099] Get:

[0100]

[0101] (3) Calculate the wavefront phase direction vector n of the wave field u = (u x , u y , u z ). The n includes n1, n2, n3, and the n1, n2, n3 respectively correspond to the wavefront phase direction vectors of the wave field u x , u y , u z . The calculation formula of n is as formula (13):

[0102]

[0103] (4) According to the wavefront phase direction vector n, the decomposition formula and decomposition operator in formula (12) are also deformed from the Poisson equation to formula (13) to obtain the new pseudo-Helmholtz decomposition formula, new decomposition operator and new Poisson equation respectively as follows:

[0104]

[0105] (5) Perform the inverse Fourier transform on equation (13) to obtain the P / S wave decomposition formula, decomposition operator and Poisson equation in the spatial domain, that is, the P / S wave decomposition formula of the anisotropic medium in the spatial domain:

[0106]

[0107] Among them, u represents the input wave field, which is a vector wave field mixed with P wave and S wave, and has three components for a three-dimensional model, u p , u s = u SH + u SV , which respectively represent the vector wave fields of P wave and S wave, respectively represent taking the spatial derivative along the x, y, z directions, w is the auxiliary wave field obtained from the input wave field u, v p , v srespectively represent the longitudinal wave and transverse wave velocities of the wave along the symmetry axis direction of the transversely isotropic medium. ε and δ are dimensionless factors representing the strength of anisotropy. The expressions of r1(x), r2(x), r3(x), and r4(x) are the same as those in formula (1). n = (n x , n y , n z ) represents the wavefront phase direction vector of the corresponding component of the input wave field.

[0108] (6) For formula (16), if the derivative term along the y direction is ignored, the anisotropic wave field separation formula in the two-dimensional case can be obtained:

[0109]

[0110] Using the wavefront phase direction vector to calculate the new pseudo-Helmholtz decomposition operator and the Poisson equation satisfied by the auxiliary wave field, that is, formulas (14) and (17), an anisotropic medium P / S wave field decomposition method with better effect than the zero-order pseudo-Helmholtz decomposition operator and smaller computational amount than the first-order pseudo-Helmholtz decomposition operator is obtained.

[0111] The present invention proposes to use the wavefront phase direction vector to implement a new "first-order pseudo-Helmholtz decomposition operator" and a new Poisson equation satisfied by the auxiliary wave field, improving the computational accuracy and efficiency of P / S wave separation, and obtaining an anisotropic medium wave field decomposition method with better effect than the zero-order pseudo-Helmholtz decomposition operator and smaller computational amount than the first-order pseudo-Helmholtz decomposition operator.

[0112] The present invention also provides a pseudo-Helmholtz decomposition device for P / S waves in an anisotropic medium, including:

[0113] The first processing unit is used to obtain the first-order pseudo-Helmholtz decomposition formula, the decomposition operator and its Poisson equation;

[0114] The second processing unit is used to transform the first-order pseudo-Helmholtz separation operator into the wavenumber domain. By calculating the wavefront phase direction vector n of the wave field u = (u x , u y , u z ), where n includes n1, n2, and n3, and n1, n2, and n3 respectively correspond to the wavefront phase direction vectors of the wave fields u x , u y , u z , to construct a new pseudo-Helmholtz decomposition formula, decomposition operator, and Poisson equation;

[0115] The third processing unit transforms the new pseudo-Helmholtz decomposition formula, decomposition operator, and auxiliary equation obtained in the wavenumber domain back to the spatial domain, and the P / S wave decomposition formula of the anisotropic medium in the spatial domain can be obtained;

[0116] The present invention also provides a computer-readable storage medium storing computer instructions, which are used to implement the pseudo Helmholtz decomposition method of longitudinal and transverse waves in anisotropic media when executed by a processor.

[0117] Specific application examples are as follows:

[0118] As Figure 2 shown, in a homogeneous anisotropic medium, the source is a 15-Hz Ricker wavelet excited by a vertical force source, the coordinates of the source are (3000 m, 3000 m), the longitudinal wave velocity v p = 3 km / s, the transverse wave v s = 1.5 km / s, the density ρ = 1.0 g / cm 3 , and the anisotropic parameters are ε = 0.2 and δ = 0.1, obtaining the velocity wave fields of the horizontal and vertical components.

[0119] Figure 3 is the P / S wave obtained by separating the wave field using the existing zero-order pseudo Helmholtz decomposition operator of Yang (2019) for Figure 2 . It can be clearly seen that the separation of longitudinal and transverse waves is incomplete, there is transverse wave energy in the longitudinal wave component, and there is longitudinal wave energy in the transverse wave component.

[0120] Figure 4 is the P / S wave obtained by separating the wave field using the new pseudo Helmholtz decomposition operator of the method of the present invention for Figure 2 . Compared with the result of Yang (2019) Figure 3 , the separation effect is significantly improved.

[0121] Example 2: In a complex anisotropic medium, using the vertical P / S wave velocity, medium density, and anisotropic parameter Hess model medium as shown in Figure 5 , the source is a 15-Hz Ricker wavelet excited by a vertical force source, and the coordinates of the source are (3000 m, 3000 m).

[0122] As Figure 6 shown, the velocity wave fields of the horizontal and vertical components obtained using the model parameters given in Figure 5 .

[0123] Figure 7 is the P wave obtained by separating the wave field using the existing zero-order pseudo Helmholtz decomposition operator of Yang (2019) for Figure 6 . It can be clearly seen that the separation of longitudinal and transverse waves is incomplete, there is obvious transverse wave energy in the longitudinal wave component, and there is also obvious longitudinal wave energy in the transverse wave component.

[0124] Figure 8 is the new pseudo Helmholtz decomposition operator using the method of the present invention for Figure 6The P-wave obtained by wavefield separation. Relative to the results of Yang (2019) Figure 7 , the separation effect is significantly improved.

[0125] Figure 9 The S-wave is obtained by using the existing zero-order pseudo-Helmholtz decomposition operator of Yang (2019) for Figure 6 wavefield separation. It can be clearly seen that the separation of longitudinal and transverse waves is incomplete, there is obvious transverse wave energy in the longitudinal wave component, and there is also obvious longitudinal wave energy in the transverse wave component.

[0126] Figure 10 The S-wave is obtained by using the new pseudo-Helmholtz decomposition operator of the above method of the present invention for Figure 6 wavefield separation. Relative to the results of Yang (2019) Figure 7 , the separation effect is significantly improved.

[0127] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some of the technical features; and these modifications or replacements 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 method for pseudo-Helmholtz decomposition of P- and S-waves in an anisotropic medium, characterized in that, including: obtaining a three-dimensional first-order pseudo Helmholtz decomposition formula, a decomposition operator, and its Poisson equation; transforming the three-dimensional first-order pseudo Helmholtz decomposition formula, the decomposition operator, and its Poisson equation into the wavenumber domain and using the wavefront phase direction to construct a new pseudo Helmholtz decomposition formula, a new decomposition operator, and its new Poisson equation, including: transforming the three-dimensional first-order pseudo Helmholtz decomposition formula, the decomposition operator, and the Poisson equation into the wavenumber domain; Calculate the wavefront phase direction vector n of the wave field u = (u x , u y , u z ), where the n includes n1, n2, n3, and the n1, n2, n3 respectively correspond to the wavefront phase direction vectors of the wave field u x , u y , u z ; representing the new pseudo Helmholtz decomposition formula, the new decomposition operator, and the new Poisson equation according to the wavefront phase direction vector n; transforming the new pseudo Helmholtz decomposition formula, the new decomposition operator, and the new auxiliary equation back to the spatial domain to obtain the longitudinal and transverse wave decomposition formula for anisotropic media in the spatial domain, and separating the longitudinal and transverse waves according to the longitudinal and transverse wave decomposition formula.

2. The pseudo Helmholtz decomposition method of longitudinal and transverse waves in an anisotropic medium according to claim 1, wherein The three-dimensional first-order pseudo Helmholtz decomposition formula, the decomposition operator, and its Poisson equation are respectively: where, u represents the input wave field, which is a vector wave field mixed with P-wave and S-wave, and has three components for a three-dimensional model, u p , u s = u SH + u SV , representing the vector wave fields of P-wave and S-wave respectively, and the S-wave includes SH-wave and SV-wave; representing the spatial derivatives along the x, y, and z directions respectively; w is the auxiliary wave field obtained from the input wave field u; v p , v s represent the longitudinal wave and transverse wave velocities of the wave along the symmetry axis direction of the transversely isotropic medium respectively, ε and δ are dimensionless factors representing the strength of anisotropy, and the expressions of r1(x), r2(x), r3(x), and r4(x) are respectively:

3. The pseudo Helmholtz decomposition method of longitudinal and transverse waves in an anisotropic medium according to claim 1, characterized in that Transforming the first-order pseudo Helmholtz decomposition formula, the decomposition operator, and the Poisson equation into the wavenumber domain, the longitudinal and transverse wave decomposition formula, the decomposition operator, and the Poisson equation under the wavenumber domain conditions are respectively: or respectively: Among them, U, U p , U s , W are the input wave fields, longitudinal wave vectors, transverse wave vectors, and auxiliary wave fields in the corresponding wavenumber domain obtained by Fourier transforming u, u p , u s , w; k x , k z represent the wavenumbers in the x and z directions; r1(x), r2(x), r3(x), r4(x) are r1, r2, r3, r4 respectively when the anisotropic parameters are constants.

4. The pseudo Helmholtz decomposition method of longitudinal and transverse waves in an anisotropic medium according to claim 1, characterized in that Using the wavefront phase direction to construct the longitudinal and transverse wave decomposition formula, the decomposition operator, and the Poisson equation under the wavenumber domain conditions into a new pseudo Helmholtz decomposition formula, a new decomposition operator, and a new auxiliary equation to obtain the separation formula for the longitudinal and transverse waves in anisotropic media in the spatial domain, including: Define the wavefront phase direction vector $\vec{n}=(n x ,n y ,n z ) And make the following substitution for the Poisson equation in Equation (2) or Equation (3): Obtained: Calculate the wavefront phase direction vector n of the wave field u = (u x , u y , u z ), where the n includes n1, n2, n3, and n1, n2, n3 respectively correspond to the wavefront phase direction vectors of the wave field u x , u y , u z . The calculation formula of n is as shown in Equation (5): According to the wavefront phase direction vector n, the decomposition formula and the decomposition operator in Equation (3) are also deformed from the Poisson equation in Equation (3) to Equation (4) to obtain the new pseudo Helmholtz decomposition formula, the new decomposition operator, and the new Poisson equation respectively: Performing an inverse Fourier transform on Equation (6) to obtain the longitudinal and transverse wave decomposition formula, the decomposition operator, and the Poisson equation in the spatial domain, that is, the longitudinal and transverse wave decomposition formula for anisotropic media in the spatial domain is: where \(u\) represents the input wave field, which is a vector wave field composed of a mixture of P-wave and S-wave, and has three components for a three-dimensional model, \(u\) p , \(u\) s = \(u\) SH + \(u\) SV , representing the vector wave fields of P-wave and S-wave respectively. representing the spatial derivatives along the \(x\), \(y\), and \(z\) directions respectively, \(w\) is an auxiliary wave field obtained from the input wave field \(u\), \(v\) p , \(v\) s represent the longitudinal wave and transverse wave velocities of the wave along the symmetry axis direction of the transversely isotropic medium respectively, \(\varepsilon\), \(\delta\) are dimensionless factors representing the strength of anisotropy, the expressions of \(r_1(x)\), \(r_2(x)\), \(r_3(x)\), \(r_4(x)\) are the same as formula (1), \(n=(n\) x , \(n\) y , \(n\) z ) represents the wavefront phase direction vector of the corresponding component of the input wave field.

5. The pseudo Helmholtz decomposition method of longitudinal and transverse waves in an anisotropic medium according to claim 4, characterized in that, also including: Ignoring the derivative term along the y direction in Equation (7), the two-dimensional anisotropic wavefield decomposition formula can be obtained: Among them, \(u\) represents the input wave field, which is a vector wave field mixed with P-wave and S-wave, \(u\) p , \(u\) s respectively represent the vector wave fields of P-wave and S-wave; respectively represent taking the spatial derivatives along the \(x\) and \(z\) directions; \(w\) is the auxiliary wave field obtained from the input wave field \(u\); \(v\) p , \(v\) s respectively represent the longitudinal wave and transverse wave velocities of the wave along the symmetry axis direction of the transversely isotropic medium, \(\varepsilon\) and \(\delta\) are dimensionless factors representing the strength of anisotropy; the expressions of \(r_1(x)\), \(r_2(x)\), \(r_3(x)\), \(r_4(x)\) are the same as those in formula (1); \(n=(n\) x , \(n\) z ) represents the wavefront phase direction vector of the corresponding component of the input wave field.

6. A pseudo-Helmholtz decomposition device for P- and S-waves in an anisotropic medium, characterized in that, including: a first processing unit for obtaining a first-order pseudo Helmholtz decomposition formula, a decomposition operator, and its Poisson equation; A second processing unit for transforming the first-order pseudo Helmholtz decomposition operator into the wavenumber domain, by calculating the wavefront phase direction vector n of the wave field u=(u x , u y , u z ), where the n includes n1, n2, n3, and the n1, n2, n3 respectively correspond to the wavefront phase direction vectors of the wave fields u x , u y , u z , and constructing a new pseudo Helmholtz decomposition formula, a new decomposition operator, and a new Poisson equation using the direction of the wavefront phase vector n; a third processing unit that transforms the new pseudo Helmholtz decomposition formula, the new decomposition operator, and the new auxiliary equation obtained in the wavenumber domain back to the spatial domain, and the longitudinal and transverse wave decomposition formula for anisotropic media in the spatial domain can be obtained.

7. A computer-readable storage medium, characterized in that, storing computer instructions, the computer instructions are used to implement the pseudo Helmholtz decomposition method for longitudinal and transverse waves in anisotropic media as described in any one of claims 1 to 5 when being executed by a processor.

8. A computer device, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the pseudo Helmholtz decomposition method for longitudinal and transverse waves in anisotropic media as described in any one of claims 1 to 5.

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