TTI medium full waveform inversion method and device based on symmetric wave equation

Abstract

The application relates to the technical field of geophysical exploration, and provides a TTI medium full waveform inversion method and device based on a symmetric wave equation, which reconstructs a wave equation by stress tensor rotation to VTI medium parameters, constructs a symmetric equation by normal stress splitting and auxiliary variable replacement, makes the adjoint equation and the forward equation consistent in mathematical structure and boundary conditions, completes forward calculation, adjoint calculation and gradient calculation in a unified numerical framework, and significantly reduces the implementation complexity of the TTI medium full waveform inversion.

Description

Technical Field

[0001] This application relates to the field of geophysical exploration technology, and in particular to a method and apparatus for full waveform inversion of TTI media based on symmetric wave equations. Background Technology

[0002] Full waveform inversion is an exploration geophysical method that can quantitatively reconstruct the elastic parameters (velocity, density, and anisotropy parameters) of the subsurface medium using all waveform information of seismic waves (including travel time, amplitude, phase, and multiple arrivals), providing a high-precision quantitative model for oil and gas reservoir identification and evaluation.

[0003] Full waveform inversion, by minimizing the mismatch between observed seismic records and theoretical forward models, can recover high-resolution parametric models of subsurface media, making it a key technology in seismic exploration. In areas with relatively simple subsurface structures, isotropic or vertical-to-lateral isotropic (VTI) assumptions can well describe wave propagation characteristics and have been widely used in industry. However, for complex sedimentary basins, inclined layered media, fault-fold zones, and areas with significant tectonic deformation, the strata often exhibit geological features such as folds and faults, causing the media's axis of symmetry to no longer be vertical but rather have a spatially varying dip angle. In these cases, the actual media better conforms to the inclined-to-lateral isotropic (TTI) model. If the VTI medium assumption is still used for modeling and inversion in these areas, it will not only lead to distortion of medium parameter characterization but may also cause gradient calculation errors, thereby reducing the accuracy and stability of wavefield simulation and full waveform inversion.

[0004] For TTI media, existing full-waveform inversion methods mainly construct wave equations from the TTI media elastic coefficient matrix in the traditional Bond transform form. While physically rigorous, this approach suffers from two prominent problems in engineering implementation: First, the elastic coefficients of TTI media exhibit complex nonlinear relationships with velocity, density, Thomsen parameters, and tilt angle, requiring repeated chain-step differentiation for gradient derivation, significantly increasing computational cost and implementation complexity. Second, the operators in the traditional TTI media wave equations generally lack symmetry, leading to inconsistencies between the adjoint state equations and the forward wave equations in both mathematical form and free surface boundary conditions, further complicating the integrated implementation of full-waveform inversion for TTI media. Specifically, due to the asymmetric form of the operator matrix, the adjoint state equations obtained by directly transposing the wave equation operators contain complex terms involving multiplying the elastic coefficients by the adjoint state variables and then performing spatial differentiation. Furthermore, the free surface boundary conditions corresponding to the adjoint state variables are quite cumbersome, further increasing the difficulty of accurate numerical implementation. It is worth noting that the mathematical structure of the TTI medium qP wave equation is basically the same as that of the elastic wave equation. Therefore, the above-mentioned difficulties also exist in the full waveform inversion of TTI medium acoustic waves.

[0005] To address the problems caused by the asymmetry of the aforementioned equations, researchers have proposed corresponding solutions for VTI media. By substituting the normal stress in the wave equation, the wave equation for VTI media can be symmetricized, allowing forward modeling and adjoint calculations to be performed within a unified numerical framework. Both numerical solutions can be completed using the same discretization scheme and program flow, significantly reducing the complexity of full waveform inversion in VTI media. However, these methods are all based on the assumption that the medium's symmetry axis is vertical (with zero tilt angle), and are only applicable to the special case where the symmetry axis direction remains unchanged. They cannot be directly extended to TTI media where the symmetry axis has a spatially varying tilt angle.

[0006] Therefore, how to achieve a unified numerical implementation of forward modeling, adjoint modeling, and gradient calculation, and reduce the engineering complexity of full waveform inversion of TTI media, has become an urgent technical challenge. Summary of the Invention

[0007] The purpose of this application is to provide a method and apparatus for full waveform inversion of TTI media based on symmetric wave equations, to achieve unified numerical implementation of forward modeling, adjoint modeling and gradient calculation, and to reduce the engineering implementation complexity of full waveform inversion of TTI media.

[0008] To achieve the above objectives, this application provides a method for full waveform inversion of TTI media based on symmetric wave equations, comprising the following steps: Acquire seismic observation data, source wavelet, and initial subsurface model of the area to be inverted; Based on the stress tensor rotation relationship between the local and global coordinate systems, the wave equation of the TTI medium is reconstructed using the elastic coefficient and tilt angle parameter of the VTI medium to simplify the parameters in the wave equation. The reconstructed TTI medium wave equation is symmetricized by splitting the normal stress term and introducing auxiliary variables for substitution, so that the symmetric form of the TTI medium wave equation with a symmetric matrix as the forward operator is constructed, and the adjoint state equation of the symmetric form of the TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions. Based on the adjoint state method, the propagating wave field and the adjoint wave field are solved under a unified numerical framework according to the symmetric TTI medium wave equation, and the gradient of the model parameters is calculated. The initial underground model is iteratively updated based on the gradient until the preset convergence condition is met, and the inversion result is output.

[0009] In the method of this application embodiment, the step of reconstructing the TTI medium wave equation based on the stress tensor rotation relationship between the local coordinate system and the global coordinate system using the VTI medium elastic coefficient and tilt angle parameter includes: Based on the rotational relationship between the Voigt scheme stress tensors in the global and local coordinate systems The elastic wave equation of TTI medium is equivalently transformed into a form expressed in terms of VTI medium elastic coefficients and tilt angle parameters;

[0010] in, For the focal term, , , These represent the horizontal and vertical components of the focal term, respectively. For sound pressure source terms, , for The corresponding scalar value, This is the Voigt stress tensor in the global coordinate system. , , , These represent the horizontal normal stress, vertical normal stress, and shear stress in the Voigt scheme of the global coordinate system. This is the stress tensor in the Voigt scheme in the local coordinate system. , , , These represent the horizontal normal stress, vertical normal stress, and shear stress in the Voigt scheme under the local coordinate system. The Bond transformation matrix is ​​related to the tilt angle of the TTI medium's symmetry axis relative to the vertical direction. , The angle of inclination of the TTI medium's axis of symmetry relative to the vertical direction. For the density of the medium, It is the identity matrix. For time, Let be the spatial differential operator matrix. , , These are the horizontal coordinates and the vertical coordinates, respectively. The velocity vector of the particle vibration. , , These represent the particle vibration velocities in the horizontal and vertical directions, respectively. Here is the matrix of elastic coefficients of VTI medium. , , , , The elastic modulus of VTI medium, For time, superscript This indicates the matrix transpose.

[0011] In the method of this application embodiment, the step of symmetricizing the reconstructed TTI medium wave equation and constructing a symmetric form TTI medium wave equation with a symmetric matrix by splitting the normal stress term and introducing auxiliary variables includes: The equation terms related to normal stress in the reconstructed TTI medium elastic wave equation are decomposed as follows:

[0012] in, The horizontal normal stress is in the local coordinate system. for The first part after splitting, for The second part after splitting, , These represent the components of the particle's vibration velocity in the horizontal and vertical directions, respectively. The angle of inclination of the TTI medium's axis of symmetry relative to the vertical direction. , , The elastic modulus of VTI medium, For sound pressure source term, For time, , These are the horizontal coordinates and the vertical coordinates, respectively. Build to satisfy Auxiliary variables and and the auxiliary variables and By substituting variables into the split equations, a symmetric form of the TTI medium elastic wave equation is constructed.

[0013] In the method of this application embodiment, the matrix form of the symmetric TTI medium elastic wave equation is:

[0014] in, For the density of the medium, It is a 2×2 identity matrix. Let be the spatial differential operator matrix. The Bond transformation matrix is ​​related to the tilt angle of the TTI medium's symmetry axis relative to the vertical direction. , To select a matrix, , The velocity vector of the particle vibration. , This is the stress-related variable vector after variable substitution. , For the focal term, , , These represent the horizontal and vertical components of the focal term, respectively. The sound pressure source term after variable substitution. , This is a diagonal matrix composed of the elastic coefficients of the VTI medium. , , and superscript This indicates the matrix transpose.

[0015] In the method of this application embodiment, the matrix form of the adjoint state equation of the symmetric TTI medium elastic wave equation is:

[0016] in, , These represent the velocity components of the accompanying wave field and the stress-related variable vector, respectively. , These are velocity-related source terms and stress-related source terms, respectively. The accompanying state equations and the forward wave equations have the same operator matrices and boundary conditions, enabling the forward and accompanying equations to be implemented using the same solver within a unified numerical framework.

[0017] In the method of this application embodiment, under acoustic approximation conditions, the symmetric form of the TTI medium elastic wave equation degenerates into the symmetric form of the TTI medium qP wave equation.

[0018] In the method of this application embodiment, the step of solving the propagating wave field and the adjoint wave field under a unified numerical framework based on the adjoint state method and the symmetric TTI medium wave equation, and calculating the gradient of the model parameters, includes: Forward modeling was performed based on the symmetric form of the TTI medium wave equation to obtain the propagating wave field and prediction data. Calculate the data residuals based on the objective function and the predicted data; The associated wave field is obtained by backpropagating the data residual according to the symmetric form of the TTI medium wave equation; According to the gradient formula Calculate the gradient of the objective function with respect to each model parameter; in, Let be the objective function. Model parameter vector The Each model parameter The maximum recording time for the observed data. For time, For the density of the medium, This is a diagonal matrix composed of the elastic coefficients of the VTI medium. The velocity component of the propagating wave field. For the velocity component of the accompanying wave field, These are the stress components of the propagating wave field. The stress components of the accompanying wave field, , They are respectively , The transpose matrix; the model parameters include the P-wave velocity. transverse wave velocity Medium density Thomsen anisotropy parameters and .

[0019] In the method of this application embodiment, the objective function is a least squares objective function:

[0020] in, For Let be the least squares objective function for the variables. For spatial regions, For predictive data, For observation data, Here are the coordinates of the receiver point. It is a unit impulse function. The coordinates are spatial coordinates; the model parameter vector includes the longitudinal wave velocity. transverse wave velocity Medium density Thomsen anisotropy parameters and .

[0021] On the other hand, embodiments of this application also provide a TTI medium full waveform inversion device based on symmetric wave equations, including: The data acquisition unit is used to acquire seismic observation data, source wavelet, and initial subsurface model of the area to be inverted. The equation reconstruction unit is used to reconstruct the wave equation of TTI medium based on the stress tensor rotation relationship between the local coordinate system and the global coordinate system, using the elastic coefficient and tilt angle parameter of VTI medium, so as to simplify the parameters in the wave equation. The equation symmetry unit is used to symmetrize the reconstructed TTI medium wave equation. By splitting the normal stress term and introducing auxiliary variables, a symmetric form of the TTI medium wave equation with a symmetric matrix as the forward operator is constructed, so that the adjoint state equation of the symmetric form of the TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions. The gradient calculation unit is used to solve the propagating wave field and the adjoint wave field under a unified numerical framework based on the adjoint state method and the symmetric TTI medium wave equation, and to calculate the gradient of the model parameters. The iterative update unit is used to iteratively update the initial underground model based on the gradient until the preset convergence condition is met, and output the inversion result.

[0022] On the other hand, embodiments of this application also provide a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the computer program, when run by the processor, executes instructions of the above-described method.

[0023] As can be seen from the technical solutions provided in the embodiments of this application above, in the embodiments of this application, after reconstructing the TTI medium wave equation, a symmetric form of the TTI medium wave equation with a symmetric matrix as the forward modeling operator can be constructed by splitting the normal stress term and introducing auxiliary variables. Since the forward modeling operator matrix is ​​a symmetric matrix, it means that the transpose of the matrix is ​​equal to itself. According to the adjoint state method, the operator matrix of the adjoint equation is mathematically the transpose of the forward modeling operator matrix, and the two are naturally completely identical. Moreover, the adjoint state equation of the symmetric form of the TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions, making the forward modeling and the adjoint equation completely consistent in terms of numerical discretization format and boundary processing method. Therefore, the difference between the forward modeling and the adjoint solution lies in the source term (the forward modeling uses physical sources, while the adjoint equation uses data residuals). In computer implementation, the same numerical solution program can be used. Only the input source term needs to be replaced from the physical source to the data residual to complete the solution of the adjoint wave field. There is no need to write a separate solution code for the adjoint equation, thereby significantly reducing the implementation complexity and engineering implementation cost. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In the drawings: Figure 1 The flowcharts of the TTI medium full waveform inversion method based on the symmetric wave equation in some embodiments of this application are shown; Figure 2 It shows Figure 1 The flowchart shown illustrates the process of constructing the reconstructed TTI medium wave equation into a symmetric form. Figure 3 It shows Figure 1The flowchart shown illustrates the method for calculating the gradient of model parameters based on the propagating wavefield and the adjoint wavefield. Figure 4 A schematic diagram of the Marmousi-II TTI model and the initial model in an exemplary embodiment of this application is shown; Figure 5 A schematic diagram of the inversion result in an exemplary embodiment of this application is shown; Figure 6 This illustration shows a schematic diagram of the model update amount in an exemplary embodiment of this application; Figure 7 This invention provides a schematic diagram comparing the longitudinal velocities of P-waves at different locations in an exemplary embodiment of this application. Figure 8 This invention provides a schematic diagram comparing the longitudinal velocities of S-waves at different locations in an exemplary embodiment of this application. Figure 9 The data of particle vibration velocity observed in an exemplary embodiment of this application are shown. x Components and predicted particle vibration velocity data x A schematic diagram comparing the waveforms between components; Figure 10 The data of particle vibration velocity observed in an exemplary embodiment of this application are shown. z Components and predicted particle vibration velocity data z A schematic diagram comparing the waveforms between components; Figure 11 A schematic diagram of the convergence curve of the normalized objective function in an exemplary embodiment of this application is shown; Figure 12 The following is a structural block diagram of a TTI medium full waveform inversion device based on a symmetric wave equation in some embodiments of this application; Figure 13 A structural block diagram of a computer device in some embodiments of this application is shown.

[0025] [Explanation of Labels in the Attached Image]

[0026] 121. Data Acquisition Unit; 122. Equation Reconstruction Unit; 123. Equation symmetry unit; 124. Gradient calculation unit; 125. Iterative update unit; 1302. Computer equipment; 1304, Processor; 1306. Memory; 1308. Drive mechanism; 1310. Input / output interface; 1312. Input devices; 1314. Output devices; 1316. Presentation equipment; 1318. Graphical User Interface; 1320. Network interface; 1322. Communication link; 1324. Communication bus. Detailed Implementation

[0027] To enable those skilled in the art to better understand the technical solutions in this application, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of this application.

[0028] It should be noted that in the embodiments of this application, the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, data stored, data displayed, etc.) involved are all information and data authorized and agreed upon by the user and fully authorized by all parties. That is, the acquisition, transmission, storage, use, and processing of data in the technical solution of this application all comply with the relevant provisions of national laws and regulations.

[0029] The basic idea of ​​full waveform inversion is to iteratively optimize model parameters so that the predicted data obtained from forward modeling and the actual observed data achieve optimal matching under a given objective function. The most commonly used method is... L The objective function defined by the 2-norm: (1) in, For model parameter vectors, For The objective function for the variable is The spatial region for which the inversion problem is to be solved is bounded by the free surface. With the ideal boundary at infinity Enclosed, The maximum recording time for the observed data. For prediction data (theoretical data calculated from the wave equation, the data type may include physical quantities such as particle vibration velocity or sound pressure). For observation data, Here are the coordinates of the receiver point. For time, It is a unit impulse function. The coordinates are spatial coordinates; the model parameter vector includes the longitudinal wave velocity. transverse wave velocity ,density Thomsen anisotropy parameters ( and ).

[0030] For the full waveform inversion of elastic waves in TTI media, the objective function (1) is constrained by the elastic wave equation of TTI media: (2) in, It is a 2×2 identity matrix; The velocity vector of the particle vibration. This is the Voigt stress tensor in the global coordinate system. , , , These represent the particle vibration velocities in the horizontal and vertical directions, respectively. , , These represent the horizontal normal stress, vertical normal stress, and shear stress in the Voigt scheme of the global coordinate system, respectively. and These are the source term (such as the concentrated force source term) and the sound pressure source term, respectively. , , , These represent the horizontal and vertical components of the focal term, respectively. for The corresponding scalar value; It is a 3×3 identity matrix; the superscript T indicates matrix transpose; and Let be the spatial differential operator matrix and the elastic coefficient matrix of the TTI medium, respectively, which can be specifically expressed as follows: (3) and (4) in, , , , , , In TTI media, in the global coordinate system, along the horizontal direction ( x The stiffness coefficient of the longitudinal wave (qP wave) propagating in the direction of (qP), the coupling stiffness coefficient between the horizontal normal stress and the vertical normal strain, the coupling stiffness coefficient between the horizontal normal stress and the shear strain, and the stiffness coefficient along the vertical direction (qP wave) zThe stiffness coefficient of longitudinal waves (qP waves) propagating in the direction of verticality, the coupling stiffness coefficient between vertical normal stress and shear strain, and the stiffness coefficient (shear modulus) of transverse waves (qS waves, polarized in the vertical plane) propagating in the horizontal or vertical direction.

[0031] It can be calculated from the VTI medium elastic coefficient matrix through Bond transformation, that is: (5) in, The Bond transformation matrix (i.e., the Bond transformation matrix related to the tilt angle of the TTI medium's symmetry axis relative to the vertical direction) can be specifically expressed as follows: (6) and (7) in, It is the angle of inclination. , , , These represent the stiffness coefficients of the longitudinal wave (qP wave) perpendicular to the axis of symmetry (horizontal direction) in VTI media, the coupling stiffness coefficient between the horizontal normal stress and the vertical normal strain, and the stiffness coefficient along the axis of symmetry (vertical direction, i.e., z The longitudinal wave (qP wave) stiffness coefficient and shear modulus (transverse wave stiffness coefficient) are given by the direction of the longitudinal wave (qP wave). Therefore, the elastic modulus of the TTI medium can be specifically expressed by the elastic modulus of the VTI medium as follows: (8) Furthermore, the elastic coefficient of the VTI medium and the model parameters The relationship is: (9) in, and These are the propagation velocities of the qP wave (quasi-P wave) and the qSV wave (quasi-SV wave) along the symmetry axis of the TI medium, respectively. and A dimensionless factor describing the anisotropic intensity of TI media; The density is the medium.

[0032] For the full waveform inversion of acoustic waves in TTI media, the objective function (1) is constrained by the TTI medium qP wave equation. The mathematical form of the equation is the same as that of the TTI medium elastic wave equation, with the only difference being the elastic coefficient, i.e.: (10) At this point, the elastic coefficient of the VTI medium and the model parameters under the acoustic approximation conditions The relationship is: (11) Furthermore, the TTI medium elastic wave equation (2) satisfies the initial conditions: (12) and boundary conditions: (13) in, Here is the normal matrix of the free surface boundary. Free surface boundary Outer normal direction vector , They are respectively In the horizontal direction (i.e.) x Direction), vertical direction (i.e.) z Directional component.

[0033] In the full waveform inversion of TTI medium, the forward propagation wave field can be numerically simulated using the finite difference method under the condition that the initial condition (12) and boundary condition (13) are satisfied, according to equation (2).

[0034] The adjoint wave field requires numerical simulation of the corresponding adjoint state equation. Through the adjoint state method, the adjoint state equation and gradient formula corresponding to equation (2) are derived as follows: (14) and (15) in, and As an accompanying source; The objective function is... Let Lagrange be the objective function; It refers to the trace of the matrix; and As an accompanying state variable, They are respectively In the horizontal direction (i.e.) x Direction), vertical direction (i.e.) z Directional component; These are respectively the accompanying horizontal normal stress component, the accompanying vertical normal stress component, and the accompanying shear stress component; and and Initial conditions are met: (16) and boundary conditions: (17) The gradient and local optimization algorithm shown in formula (15) can be used to iteratively update the initial model parameters, thereby realizing the inversion of the underground model parameters, that is, to find the specific values ​​of the underground model parameters, including the P-wave and S-wave velocities and anisotropic parameters.

[0035] By comparing the mathematical structures of the TTI medium wave equation (2) and its adjoint state equation (15), it can be found that the differential operator matrices on the left-hand side of the two equations satisfy the transpose relation. Since the operator matrix of the TTI medium elastic wave equation (2) is asymmetric, the mathematical structure of the adjoint state equation is not consistent with that of the wave equation, which significantly increases the complexity of numerical implementation. In particular, there are terms in the adjoint state equation where the elastic coefficient is multiplied by the adjoint state variable and then spatially differentiated. The handling of such terms is quite cumbersome in numerical implementation. In addition, the free surface boundary conditions corresponding to the adjoint state variables are quite complex, which further increases the difficulty of accurate numerical implementation. On the other hand, the algebraic relationship between the TTI medium elastic wave equation and the model parameters based on the Bond transform method is very complex. The calculation of the gradient of the model parameters is difficult to obtain directly. It usually requires multiple expansions and cumulative calculations using the chain rule, which significantly increases the computational cost and implementation complexity. The mathematical structures of the TTI medium qP wave equation and the elastic wave equation (2) are basically consistent. Therefore, the implementation of full waveform inversion based on the TTI medium elastic wave equation also faces the above difficulties.

[0036] For the full waveform inversion of elastic waves in VTI media, the objective function (1) is an optimization problem established under the constraints of the VTI medium elastic wave equation. Since VTI media is a special case of TTI media under the condition that the tilt angle of the axis of symmetry is always 0, fixing the tilt angle parameter in the TTI medium elastic wave equation (2) to zero will naturally degenerate into the VTI medium elastic wave equation, the expression of which is: (18) As can be seen from equation (18), the operator matrix of the VTI medium elastic wave equation also has an asymmetric structure. Therefore, the traditional VTI medium elastic wave full waveform inversion faces a similar problem to that of the TTI medium in numerical implementation, namely, its adjoint state equation is inconsistent with the forward wave equation in terms of mathematical structure and free surface boundary conditions, thus increasing the difficulty of numerical implementation of the adjoint equation. To overcome this problem, the VTI medium wave equation symmetry method can be adopted. This method achieves the symmetry of the VTI medium wave equation by substituting variables for the normal stress term in the wave equation. The symmetric form of the VTI medium elastic wave equation is: (19) in, , , The superscript -1 indicates the inverse matrix. Similarly, under the acoustic approximation, the symmetric form of the VTI medium qP wave equation can be expressed as: (20) The coefficients in this equation can be calculated using the VTI medium elastic coefficients (11) under the acoustic approximation: (twenty one) Since the differential operator matrices on the left-hand sides of the wave equation and the adjoint state equation are transposes of each other, when the wave equation has a symmetric structure, its corresponding adjoint state equation is consistent with the forward wave equation in terms of mathematical structure and free surface boundary conditions. Therefore, forward modeling and adjoint calculations can be implemented within a unified numerical framework, and both can be solved numerically using the same discretization scheme and program flow, thus significantly reducing the difficulty of implementing the adjoint state equation and the overall implementation complexity.

[0037] Existing full-waveform inversion methods for VTI media are all based on the assumption that the medium's axis of symmetry is vertical (with zero dip angle), and are only applicable to the special case where the direction of the axis of symmetry remains unchanged. However, actual subsurface media are affected by factors such as folds, faults, and tectonic deformation, and their axes of symmetry usually have spatially varying dip angles, which are more consistent with the transversely isotropic (TTI) medium model. Therefore, VTI medium models are difficult to accurately describe the anisotropic characteristics of complex subsurface media. If VTI media are still used for modeling and inversion, it will not only lead to distortion of medium parameter characterization, but may also cause gradient calculation errors, thereby reducing the accuracy and stability of wavefield simulation and full-waveform inversion.

[0038] To address the problems existing in TTI medium full waveform inversion methods, such as asymmetric equations, difficulty in unifying the adjoint state equation and forward equation, reliance on complex chain derivatives for gradient derivation, cumbersome implementation of free surface boundaries, and high engineering costs, this application proposes a TTI medium full waveform inversion method and apparatus based on symmetric wave equations (i.e., wave equations in symmetric form). This scheme aims to achieve unified numerical implementation of forward modeling, adjoint equations, and gradient calculations, and reduce the implementation complexity of TTI medium full waveform inversion.

[0039] This application provides a full waveform inversion method for TTI media based on symmetric wave equations, which can be applied to computer equipment. (See reference...) Figure 1 As shown in some embodiments of this application, the TTI medium full waveform inversion method based on the symmetric wave equation may include the following steps: Step 101: Obtain earthquake observation data, source wavelet, and initial underground model of the area to be inverted.

[0040] Step 102: Based on the stress tensor rotation relationship between the local coordinate system and the global coordinate system, reconstruct the wave equation of the TTI medium using the VTI medium elastic coefficient and tilt angle parameter to simplify the parameters in the wave equation.

[0041] Step 103: The reconstructed TTI medium wave equation is symmetricized by splitting the normal stress term and introducing auxiliary variables for substitution to construct a symmetric form of the TTI medium wave equation with a symmetric matrix as the forward operator, so that the adjoint state equation of the symmetric form of the TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions.

[0042] Step 104: Based on the adjoint state method, solve the propagating wave field and the adjoint wave field under a unified numerical framework according to the symmetric TTI medium wave equation, and calculate the gradient of the model parameters.

[0043] Step 105: Iteratively update the initial underground model based on the gradient until the preset convergence condition is met, and output the inversion result.

[0044] In this embodiment, after reconstructing the TTI medium wave equation, a symmetric form of the TTI medium wave equation with a symmetric matrix as the forward modeling operator can be constructed by splitting the normal stress term and introducing auxiliary variables. Since the forward modeling operator matrix is ​​a symmetric matrix, it means that the transpose of the matrix is ​​equal to itself. According to the adjoint state method, the operator matrix of the adjoint equation is mathematically the transpose of the forward modeling operator matrix, and the two are naturally identical. Moreover, the adjoint state equation of the symmetric form of the TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions, making the forward modeling and adjoint equations completely consistent in terms of numerical discretization format and boundary treatment. Therefore, the difference between the forward modeling and adjoint equation solutions lies in the source term (physical source in the forward modeling, data residual in the adjoint equation). In computer implementation, the same numerical solution program can be used. Only the input source term needs to be replaced from the physical source to the data residual to complete the solution of the adjoint wave field. There is no need to write separate solution code for the adjoint equation, thereby significantly reducing the implementation complexity and engineering implementation cost.

[0045] Seismic observation data is acquired by arrays of geophones deployed on the surface or in wells, including particle vibration velocity components. , And / or sound pressure data. The source wavelet is a time function of the energy source that generates seismic waves, typically a Ricker wavelet or an equivalent source signal extracted from actual source records. The initial model can be a smooth model; shear wave velocity and density can be obtained from P-wave velocity using empirical formulas (such as Gardner's formula), or from well logging data or prior geological information. The initial subsurface model must include at least P-wave velocity, shear wave velocity, density, and Thomsen anisotropy parameters (…). and and inclination angle Spatial distribution; where, tilt angle The angle of inclination of the TTI medium's axis of symmetry relative to the vertical direction.

[0046] In some embodiments of this application, the global coordinate system may refer to a Cartesian coordinate system fixed to the Earth's surface, typically using... x The axis indicates the horizontal direction. z The axis represents the vertically downward direction. A local coordinate system can refer to a coordinate system fixed along the direction of the medium's axis of symmetry. z The axis is aligned with the axis of symmetry of the TTI medium.

[0047] In some embodiments of this application, the stress tensor rotation relationship refers to the coordinate transformation relationship between the Voigt format stress tensor in the global coordinate system and the local coordinate system, which can be implemented by the Bond transformation matrix.

[0048] In some embodiments of this application, reconstructing the wave equation of TTI medium based on the stress tensor rotation relationship between the local coordinate system and the global coordinate system using the VTI medium elastic coefficient and tilt angle parameter may include: Based on the rotational relationship between the Voigt scheme stress tensors in the global and local coordinate systems The elastic wave equation for TTI medium is equivalently transformed into a form expressed in terms of VTI medium elastic coefficients and tilt angle parameters:

[0049] in, This is the Voigt stress tensor in the local coordinate system. .

[0050] Through this rotational relationship, the global TTI medium elastic coefficient matrix Replaced with VTI medium elastic coefficient matrix With tilt angle The combination avoids directly expanding and differentiating the global TTI medium elastic coefficient matrix after Bond transformation, thus helping to reduce implementation costs.

[0051] refer to Figure 2 As shown, in some embodiments of this application, the reconstructed TTI medium wave equation is symmetricized, and a symmetric form of the TTI medium wave equation with a symmetric matrix is ​​constructed by splitting the normal stress term and introducing auxiliary variables. This may include the following steps: Step 201: Decompose the equation terms related to normal stress in the reconstructed TTI medium elastic wave equation.

[0052] In some embodiments of this application, the equation terms related to normal stress in the reconstructed TTI medium elastic wave equation can be decomposed according to the following formula:

[0053] in, The horizontal normal stress is in the local coordinate system. for The first part after splitting, for The second part after splitting, , These are the horizontal coordinates and the vertical coordinates, respectively.

[0054] Step 202: Construct auxiliary variables that meet preset conditions. and and the auxiliary variables and By substituting variables into the split equations, a symmetric form of the TTI medium elastic wave equation is constructed.

[0055] In some embodiments of this application, the preset conditions can be: .

[0056] In some embodiments of this application, the symmetric form of the TTI medium elastic wave equation can be expressed as:

[0057] Furthermore, the matrix form of this symmetric TTI medium elastic wave equation can be expressed as:

[0058] in, To select a matrix, , The four-component stress-related variable vector after variable substitution Map back to the three-component Voigt-style stress tensor The stress-related variable vector after variable substitution and , The sound pressure source term after variable substitution. , It is a diagonal matrix composed of the elastic coefficients of the VTI medium, i.e. ,in , and .

[0059] Furthermore, the wave field variables in the matrix form of the symmetric TTI medium elastic wave equation satisfy the initial conditions. and boundary conditions .

[0060] In the left-hand operator matrix of this matrix equation, the upper right block With the bottom left block They are transposes of each other, top left block and the bottom right block All of them are symmetric matrices (diagonal matrices), therefore the entire operator matrix is ​​a symmetric matrix.

[0061] In some embodiments of this application, the adjoint state equation of the above-described symmetric form of the TTI medium elastic wave equation can be expressed as: ; Furthermore, the matrix form of the adjoint state equation of the symmetric TTI medium elastic wave equation can be expressed as:

[0062] in, , These represent the velocity components of the accompanying wave field and the stress-related variable vector, respectively. , These are velocity-related source terms and stress-related source terms, respectively. , The termination conditions are met. and boundary conditions .

[0063] The accompanying source corresponding to the above wave equation after variable substitution The specific mathematical expression is:

[0064] The aforementioned adjoint state equation shares the same left-hand operator matrix and boundary conditions as the forward wave equation, providing a direct mathematical basis for realizing both forward and adjoint wave equations within a unified numerical framework. This allows both equations to be implemented using the same solver within the same numerical framework. The adjoint state equation refers to the partial differential equation satisfied by the adjoint wave field, derived from the gradient of the objective function with respect to the model parameters using the adjoint state method (a mathematical method for calculating the gradient of the objective function with respect to the model parameters by solving the adjoint equation). Its left-hand operator matrix is ​​the transpose of the forward operator matrix.

[0065] In some embodiments of this application, under acoustic approximation conditions, i.e. The symmetric form of the TTI medium elastic wave equation can be directly degenerated into the symmetric form of the TTI medium qP wave equation. The computational complexity of the qP wave equation is significantly lower than that of the elastic wave equation, making it suitable for large-scale exploration scenarios that only require P-wave information. The symmetric form of the qP wave equation in this application also has the characteristic that the forward modeling operator matrix is ​​a symmetric matrix. Its adjoint state equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions, thus enabling the solution of both forward modeling and adjoint wave equations within a unified numerical framework.

[0066] Under the acoustic approximation, the symmetric form of the qP wave equation for the TTI medium can be expressed as:

[0067] The coefficients in this equation are the VTI medium elastic coefficients under the acoustic approximation shown in the equation.

[0068] refer to Figure 3 As shown, in some embodiments of this application, based on the adjoint state method, solving the propagating wave field and adjoint wave field under a unified numerical framework according to the symmetric form of the TTI medium wave equation, and calculating the gradient of the model parameters, may include the following steps: Step 301: Perform forward modeling based on the symmetrical TTI medium wave equation to obtain the forward propagation wave field and prediction data.

[0069] In some embodiments of this application, forward modeling can be performed using the finite difference method while satisfying initial conditions ( ) and boundary conditions ( This is done under the following circumstances.

[0070] Step 302: Calculate the data residuals based on the objective function and the predicted data.

[0071] In some embodiments of this application, data residuals can be calculated based on the objective function and the predicted data. - And use the data residual as a concomitant source.

[0072] Step 303: Backpropagate the data residual according to the symmetrical TTI medium wave equation to obtain the associated wave field.

[0073] In some embodiments of this application, the accompanying wave field is obtained by using the backpropagation data residuals of the TTI medium wave equation in a symmetric form. and The solution to the adjoint wave field satisfies the termination condition ( ) and boundary conditions ( ).

[0074] Step 304: According to the gradient formula Calculate the gradient of the objective function with respect to each model parameter.

[0075] in, Let be the objective function. For the first Each model parameter The velocity component of the propagating wave field. For the velocity component of the accompanying wave field, These are the stress components of the propagating wave field. The stress components of the accompanying wave field, This is a diagonal matrix composed of the elastic coefficients of the VTI medium. , They are respectively , The transpose matrix; the model parameters include the P-wave velocity. transverse wave velocity ,density Thomsen anisotropy parameters ( and The dip angle parameter can be obtained by measuring the dip angle of the layer in the velocity model. During the inversion process, it can be fixed to a known value, or the dip angle model can be updated by measuring the dip angle of the layer in the updated velocity model.

[0076] from Figure 3 As can be seen from the illustrated embodiment, since the forward operator matrix is ​​a symmetric matrix, the solutions for the propagating wavefield and the adjoint wavefield use the exact same numerical solver, differing only in the source terms, thus significantly reducing the implementation complexity of gradient calculation. Furthermore, because the stress tensor rotation reconstruction avoids the complex chain derivatives introduced by the Bond transformation, the gradient formula no longer contains... For complex terms, gradient calculation is more direct and efficient.

[0077] In some embodiments of this application, the objective function can be a least squares objective function:

[0078] in, For model parameter vectors, For Let be the least squares objective function for the variables. At the detector point A generalized function that takes a value of infinity at one location and zero at other locations is used to extract the wave field value at the location of a discrete receiver point from a continuous wave field.

[0079] In some embodiments of this application, the preset convergence condition can be that the gradient norm meets a threshold or reaches the maximum number of iterations; in this case, when the objective function converges, the gradient norm meets the threshold or reaches the maximum number of iterations, the inversion result is output.

[0080] The symmetry transformation and full waveform inversion of the two-dimensional TTI medium elastic wave equation are illustrated in some embodiments of this application. However, the method described in the embodiments of this application is also applicable to the symmetry transformation and full waveform inversion of the three-dimensional TTI medium elastic wave equation. The implementation steps for the symmetry transformation and full waveform inversion of the three-dimensional TTI medium elastic wave equation are basically the same as those for the two-dimensional TTI medium elastic wave equation. In this scenario, the matrix form of the symmetric TTI medium elastic wave equation is:

[0081] Among them, superscript Indicates matrix transpose. It is a 3×3 identity matrix. Let be the spatial differential operator matrix. , To select a matrix, , The Bond transformation matrix is ​​related to the tilt angle of the symmetry axis of the three-dimensional TTI medium relative to the vertical direction: , The azimuth angle of the TTI medium's axis of symmetry relative to the vertical direction. The velocity vector of the particle vibration. , This is the stress-related variable vector after variable substitution. ,satisfy , This is the three-dimensional stress tensor in Voigt scheme in the local coordinate system. , For the focal term, , , , These are the components of the source term along the three coordinate axes. The sound pressure source term after variable substitution. , It is a diagonal matrix composed of the elastic coefficients of a three-dimensional VTI medium. , , and , , , , , The elastic coefficient of the three-dimensional VTI medium.

[0082] The technical effects of this application are further illustrated below with reference to numerical embodiments.

[0083] like Figure 4The Marmousi-II model shown was used for a numerical experiment to perform full waveform inversion of elastic waves in TTI media. The dip parameters in the model were obtained through stratigraphic dip measurements, and the density parameters were calculated from the P-wave velocity model using Gardner's empirical formula. The initial velocity model used for the full waveform inversion was obtained from the actual model through 15 two-dimensional mean smoothing processes with a window size of 21×21. Among these, in... Figure 4 In the diagram, (a) represents the true longitudinal wave velocity. (c) represents the actual shear wave velocity. (e) represents the true anisotropy parameters. (g) represents the true anisotropy parameters. (i) represents the true dip angle. ; Figure 4 Parts (b), (d), (f), (h), and (j) shown on the right are all initial models obtained after performing two-dimensional mean smoothing on the corresponding real models on the left.

[0084] The full waveform inversion employs a multi-scale strategy, performed sequentially across six frequency bands: 0–2 Hz, 0–4 Hz, 0–6 Hz, 0–8 Hz, 0–10 Hz, and the full frequency band. Data for each frequency band is obtained through low-pass filtering with the corresponding cutoff frequency as the passband edge. Each frequency band is iterated 10 times, for a total of 60 iterations. In this embodiment, only the velocity and anisotropy parameters are updated, while the tilt parameter remains unchanged; in other embodiments, the tilt parameter can also be included in the joint update. The density parameter can be synchronously calculated from the P-wave velocity model updated in each iteration using the Gardner formula.

[0085] The inversion results and model update amounts are as follows: Figure 5 and Figure 6 As shown. Figure 5 In the diagram, (a) represents the longitudinal wave velocity obtained after inversion. (b) represents the transverse wave velocity obtained after inversion. (c) represents the anisotropy parameters obtained after inversion. (d) represents the anisotropy parameters obtained after inversion. . Figure 6 In the diagram, (a) represents the P-wave velocity increment (update) obtained after the inversion. (b) represents the shear wave velocity increment obtained after inversion. (c) represents the anisotropy parameter increments obtained after the inversion. (d) represents the anisotropy parameter increments obtained after the inversion. . Figure 5 and Figure 6The results show that after multi-scale iterative updates, the P-wave and S-wave velocity models are significantly improved, and the inversion results are highly consistent with the actual models.

[0086] Figure 7 and Figure 8 The P-wave velocities (i.e., longitudinal wave velocities) at three different lateral locations (approximately 2 km, 4 km, and 6 km) are given. ) and S-wave velocity (i.e., transverse wave velocity) The longitudinal section comparison results of ). Among them, Figure 7 and Figure 8 The solid line represents the true value, the dashed line represents the initial value, and the dotted-dash line represents the inverted value. From... Figure 7 and Figure 8 As can be seen, although the anisotropic parameter is a weakly sensitive parameter and its inversion result is difficult to fully recover the true value, it still shows a significant improvement compared to the initial model.

[0087] Furthermore, Figure 9 The observed particle vibration velocity data are shown. x Components and predicted particle vibration velocity data x A schematic diagram comparing the waveforms between components; Figure 10 The observed particle vibration velocity data are shown. z Components and predicted particle vibration velocity data z A waveform comparison diagram between components; such as Figure 9 and Figure 10 As shown, by performing forward modeling using the initial model and the inversion model and comparing it with the observed data, it can be seen that the matching degree between the predicted data and the observed data in terms of travel time and phase characteristics is significantly improved after inversion. Specifically, Figure 9 and Figure 10 The horizontal axis "Offset" represents the offset distance, which refers to the epicenter (where the epicenter is located). x The horizontal distance between the detector and the detector at point = 4.625; Figure 9 and Figure 10 The solid line represents observed data, and the dashed line represents predicted data.

[0088] Figure 11 The convergence curve of the normalized objective function with the number of iterations is shown. Figure 11 The horizontal axis represents the number of iterations, and the vertical axis represents the normalized objective function value; see also Figure 11 It can be seen that the curve drops rapidly in the early stage of the inversion, indicating that the gradient direction is correct and the model quickly approaches the true solution; after entering the middle stage, the rate of decline slows down but remains stable without oscillation or rebound; finally, it tends to flatten and converges to a lower value, indicating that the method implemented in this application has good numerical stability and convergence, and can be reliably applied to the full waveform inversion of actual TTI media.

[0089] Although the process described above includes multiple operations that occur in a specific order, it should be clearly understood that these processes may include more or fewer operations that can be executed sequentially or in parallel (e.g., using parallel processors or a multithreaded environment).

[0090] Corresponding to the aforementioned TTI medium full waveform inversion method based on symmetric wave equations, this application also provides a TTI medium full waveform inversion device based on symmetric wave equations, referencing... Figure 12 As shown, in some embodiments of this application, the TTI medium full waveform inversion device based on the symmetric wave equation may include: Data acquisition unit 121 is used to acquire seismic observation data, source wavelet, and initial subsurface model of the area to be inverted; Equation reconstruction unit 122 is used to reconstruct the wave equation of TTI medium based on the stress tensor rotation relationship between the local coordinate system and the global coordinate system, using the elastic coefficient and tilt angle parameter of VTI medium, so as to simplify the parameters in the wave equation. The equation symmetry unit 123 is used to symmetricize the reconstructed TTI medium wave equation and construct a symmetric TTI medium wave equation with a symmetric matrix as the forward operator by splitting the normal stress term and introducing auxiliary variables, so that the adjoint state equation of the symmetric TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions. The gradient calculation unit 124 is used to solve the propagating wave field and the adjoint wave field under a unified numerical framework based on the adjoint state method and the symmetric TTI medium wave equation, and to calculate the gradient of the model parameters. The iterative update unit 125 is used to iteratively update the initial underground model based on the gradient until the preset convergence condition is met, and output the inversion result.

[0091] For ease of description, the above devices are described separately by function as various units. Of course, in implementing this application, the functions of each unit can be implemented in one or more software and / or hardware.

[0092] Embodiments of this application also provide a computer device. For example... Figure 13As shown, in some embodiments of this application, the computer device 1302 may include one or more processors 1304, such as one or more central processing units (CPUs) or graphics processing units (GPUs), each of which may implement one or more hardware threads. The computer device 1302 may also include any memory 1306 for storing any kind of information such as code, settings, data, etc. In one specific embodiment, a computer program on the memory 1306 and executable on the processor 1304, when run by the processor 1304, can execute instructions of the TTI medium full waveform inversion method based on the symmetric wave equation described in any of the above embodiments. Non-limitingly, for example, the memory 1306 may include any type of RAM, any type of ROM, flash memory device, hard disk, optical disk, etc. More generally, any memory can use any technology to store information. Further, any memory can provide volatile or non-volatile retention of information. Further, any memory may represent a fixed or removable component of the computer device 1302. In one scenario, when processor 1304 executes associated instructions stored in any memory or combination of memories, computer device 1302 can perform any operation of the associated instructions. Computer device 1302 also includes one or more drive mechanisms 1308 for interacting with any memory, such as hard disk drive mechanisms, optical disk drive mechanisms, etc.

[0093] Computer device 1302 may also include an input / output interface 1310 (I / O) for receiving various inputs (via input device 1312) and providing various outputs (via output device 1314). A specific output mechanism may include a presentation device 1316 and an associated graphical user interface 1318 (GUI). In other embodiments, the input / output interface 1310 (I / O), input device 1312, and output device 1314 may be omitted, and the device may function solely as a computer device within a network. Computer device 1302 may also include one or more network interfaces 1320 for exchanging data with other devices via one or more communication links 1322. One or more communication buses 1324 couple the components described above together.

[0094] Communication link 1322 can be implemented in any way, such as via a local area network, a wide area network (e.g., the Internet), a point-to-point connection, or any combination thereof. Communication link 1322 may include any combination of hardwired links, wireless links, routers, gateway functions, name servers, etc., governed by any protocol or combination of protocols.

[0095] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), computer-readable storage media, and computer program products according to some embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processor to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processor, create a machine for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0096] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processor to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0097] These computer program instructions may also be loaded onto a computer or other programmable data processor, causing a series of operational steps to be performed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable device for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0098] In a typical configuration, a computer device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.

[0099] Memory may include non-persistent storage in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.

[0100] Computer-readable media, including both permanent and non-permanent, removable and non-removable media, can store information using any method or technology. Information can be computer-readable instructions, data structures, program modules, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by computer equipment. As defined in this application, computer-readable media does not include transient media, such as modulated data signals and carrier waves.

[0101] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, embodiments of this application can take the form of entirely hardware embodiments, entirely software embodiments, or embodiments combining software and hardware aspects. Furthermore, embodiments of this application can take the form of computer program products implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0102] The embodiments of this application can be described in the general context of computer-executable instructions, such as program modules, that are executed by a computer. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform a specific task or implement a specific abstract data type. The embodiments of this application can also be practiced in distributed computing environments where tasks are performed by remote processors connected via a communication network. In distributed computing environments, program modules can reside in local and remote computer storage media, including storage devices.

[0103] It should also be understood that, in the embodiments of this application, the term "and / or" is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, and B existing alone. Additionally, the character " / " in this document generally indicates that the preceding and following related objects have an "or" relationship.

[0104] The various embodiments in this application are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. In particular, the system embodiments are basically similar to the method embodiments, so the description is relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

[0105] In the description of this application, the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the embodiments of this application. In this application, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in a suitable manner in any one or more embodiments or examples. Furthermore, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this application, as well as the features of different embodiments or examples.

[0106] The above description is merely an embodiment of this application and is not intended to limit the scope of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of the claims of this application.

Claims

1. A method for full waveform inversion of TTI media based on symmetric wave equations, characterized in that, Includes the following steps: Acquire seismic observation data, source wavelet, and initial subsurface model of the area to be inverted; Based on the stress tensor rotation relationship between the local and global coordinate systems, the wave equation of the TTI medium is reconstructed using the elastic coefficient and tilt angle parameter of the VTI medium to simplify the parameters in the wave equation. The reconstructed TTI medium wave equation is symmetricized by splitting the normal stress term and introducing auxiliary variables for substitution, so that the symmetric form of the TTI medium wave equation with a symmetric matrix as the forward operator is constructed, and the adjoint state equation of the symmetric form of the TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions. Based on the adjoint state method, the propagating wave field and the adjoint wave field are solved under a unified numerical framework according to the symmetric TTI medium wave equation, and the gradient of the model parameters is calculated. The initial underground model is iteratively updated based on the gradient until the preset convergence condition is met, and the inversion result is output.

2. The method according to claim 1, characterized in that, The method for reconstructing the wave equation of TTI medium based on the stress tensor rotation relationship between the local and global coordinate systems, using the VTI medium elastic coefficient and tilt angle parameter, includes: Based on the rotational relationship between the Voigt scheme stress tensors in the global and local coordinate systems The elastic wave equation of TTI medium is equivalently transformed into a form expressed in terms of VTI medium elastic coefficients and tilt angle parameters: in, For the focal term, , , These represent the horizontal and vertical components of the focal term, respectively. For sound pressure source terms, , for The corresponding scalar value, This is the Voigt stress tensor in the global coordinate system. , , , These represent the horizontal normal stress, vertical normal stress, and shear stress in the Voigt scheme of the global coordinate system. This is the stress tensor in the Voigt scheme in the local coordinate system. , , , These represent the horizontal normal stress, vertical normal stress, and shear stress in the Voigt scheme under the local coordinate system. The Bond transformation matrix is ​​related to the tilt angle of the TTI medium's symmetry axis relative to the vertical direction. , The angle of inclination of the TTI medium's axis of symmetry relative to the vertical direction. For the density of the medium, It is the identity matrix. For time, Let be the spatial differential operator matrix. , , These are the horizontal coordinates and the vertical coordinates, respectively. The velocity vector of the particle vibration. , , These represent the particle vibration velocities in the horizontal and vertical directions, respectively. Here is the matrix of elastic coefficients of VTI medium. , , , , The elastic modulus of VTI medium, For time, superscript This indicates the matrix transpose.

3. The method according to claim 1, characterized in that, The process of symmetricizing the reconstructed TTI medium wave equation and constructing a symmetric form TTI medium wave equation with a symmetric matrix by splitting the normal stress term and introducing auxiliary variables includes: The equation terms related to normal stress in the reconstructed TTI medium elastic wave equation are decomposed as follows: in, The horizontal normal stress in the local coordinate system. for The first part after splitting, for The second part after splitting, , These represent the components of the particle's vibration velocity in the horizontal and vertical directions, respectively. The angle of inclination of the TTI medium's axis of symmetry relative to the vertical direction. , , The elastic modulus of VTI medium, For sound pressure source term, For time, , These are the horizontal coordinates and the vertical coordinates, respectively. Build to satisfy Auxiliary variables and and the auxiliary variables and By substituting variables into the split equations, a symmetric form of the TTI medium elastic wave equation is constructed.

4. The method according to claim 3, characterized in that, The matrix form of the symmetric TTI medium elastic wave equation is as follows: in, For the density of the medium, It is a 2×2 identity matrix. Let be the spatial differential operator matrix. The Bond transformation matrix is ​​related to the tilt angle of the TTI medium's symmetry axis relative to the vertical direction. , To select a matrix, , The velocity vector of the particle vibration. , This is the stress-related variable vector after variable substitution. , For the focal term, , , These represent the horizontal and vertical components of the focal term, respectively. The sound pressure source term after variable substitution. , This is a diagonal matrix composed of the elastic coefficients of the VTI medium. , , and superscript This indicates the matrix transpose.

5. The method according to claim 4, characterized in that, The matrix form of the adjoint state equation of the symmetric TTI medium elastic wave equation is as follows: in, , These represent the velocity components of the accompanying wave field and the stress-related variable vector, respectively. , These are velocity-related source terms and stress-related source terms, respectively. The accompanying state equations and the forward wave equations have the same operator matrices and boundary conditions, enabling the forward and accompanying equations to be implemented using the same solver within a unified numerical framework.

6. The method according to claim 1, characterized in that, Under acoustic approximation conditions, the symmetric form of the TTI medium elastic wave equation degenerates into the symmetric form of the TTI medium qP wave equation.

7. The method according to claim 1, characterized in that, The method based on the adjoint state, which solves for the propagating wave field and the adjoint wave field within a unified numerical framework according to the symmetric TTI medium wave equation, and calculates the gradient of the model parameters, includes: Forward modeling was performed based on the symmetric form of the TTI medium wave equation to obtain the propagating wave field and prediction data. Calculate the data residuals based on the objective function and the predicted data; The associated wave field is obtained by backpropagating the data residual according to the symmetric form of the TTI medium wave equation; According to the gradient formula Calculate the gradient of the objective function with respect to each model parameter; in, Let be the objective function. Model parameter vector The Each model parameter The maximum recording time for the observed data. For time, For the density of the medium, This is a diagonal matrix composed of the elastic coefficients of the VTI medium. The velocity component of the propagating wave field. For the velocity component of the accompanying wave field, These are the stress components of the propagating wave field. The stress components of the accompanying wave field, , They are respectively , The transpose matrix; the model parameters include the P-wave velocity. transverse wave velocity Medium density Thomsen anisotropy parameters and .

8. The method according to claim 7, characterized in that, The objective function is a least squares objective function: in, For Let be the least squares objective function for the variables. For spatial regions, For predictive data, For observation data, Here are the coordinates of the receiver point. It is a unit impulse function. The coordinates are spatial coordinates; the model parameter vector includes the longitudinal wave velocity. transverse wave velocity Medium density Thomsen anisotropy parameters and .

9. A TTI medium full waveform inversion device based on symmetric wave equations, characterized in that, include: The data acquisition unit is used to acquire seismic observation data, source wavelet, and initial subsurface model of the area to be inverted. The equation reconstruction unit is used to reconstruct the wave equation of TTI medium based on the stress tensor rotation relationship between the local coordinate system and the global coordinate system, using the elastic coefficient and tilt angle parameter of VTI medium, so as to simplify the parameters in the wave equation. The equation symmetry unit is used to symmetrize the reconstructed TTI medium wave equation and construct a symmetric TTI medium wave equation with a symmetric matrix as the forward operator by splitting the normal stress term and introducing auxiliary variables, so that the adjoint state equation of the symmetric TTI medium wave equation is consistent with the forward wave equation in terms of mathematical structure and boundary conditions. The gradient calculation unit is used to solve the propagating wave field and the adjoint wave field under a unified numerical framework based on the adjoint state method and the symmetric TTI medium wave equation, and to calculate the gradient of the model parameters. The iterative update unit is used to iteratively update the initial underground model based on the gradient until the preset convergence condition is met, and output the inversion result.

10. A computer device comprising a memory, a processor, and a computer program stored in the memory, characterized in that, When the computer program is run by the processor, it executes the instructions of the method according to any one of claims 1-8.

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