Elastic wave field separation method, device and medium based on Helmholtz decomposition
By combining scalar Poisson equations and smooth extension technology, the problems of high computational cost and artifacts in the Helmholtz decomposition method are solved, and efficient elastic wavefield separation is achieved, especially in three-dimensional problems, which reduces the computational cost and eliminates boundary artifacts.
Patent Information
- Application Number
- CN202310191619.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-03-02
- Publication Date
- 2025-07-25
- Estimated Expiration
- 2043-03-02
AI Technical Summary
The existing elastic wave field separation method based on Helmholtz decomposition is computationally cost-effective and is prone to artifacts in the separated P wave field boundaries and S wave field boundaries, especially in three-dimensional problems.
The wavefield separation method based on scalar Poisson equation is adopted, combining the connection between the Laplace operator in the spatial domain and the wavenumber domain, and using the smooth extension technology of the truncation function to deal with the boundary problem, the smooth extension of the input wavefield divergence is calculated through the finite difference method, and the absorption boundary is retained to avoid artifacts.
The calculation cost is significantly reduced, and artifacts are eliminated in the separated P wave field boundaries and S wave field boundaries, improving the calculation efficiency.
Smart Images
Figure CN116165704B_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of elastic wave separation, and particularly relates to an elastic wave field separation method, device and medium based on Helmholtz decomposition. Background Art
[0002] Seismic waves are elastic waves, which include P-waves and S-waves. These two types of waves are coupled during propagation, making it difficult to directly use these two waves for imaging and inversion. Therefore, effectively separating S-waves and P-waves from the coupled elastic wave field is of great significance in elastic medium reverse time migration and full waveform inversion, which helps to eliminate wave mode crosstalk to improve the quality of imaging and inversion.
[0003] In isotropic media, there are currently three types of elastic wave separation methods. The first type of method is based on directly applying divergence and curl operators to the coupled elastic wave field. Due to the polarity reversal problem and some other drawbacks, such as amplitude and phase distortion, etc., the application of this method is limited. The second type of method is to perform vector wave field decomposition by decoupling the first-order elastic wave equation and introducing an auxiliary P-wave equation. The S-wave is obtained by directly subtracting the P-wave from the coupled input wave domain. In addition, the separated wave field retains its original amplitude and phase. One drawback of this method is that if the velocity model is not smooth, artifacts will appear at sharp interfaces. The third type of method is based on Helmholtz decomposition. Helmholtz decomposition was first proposed by Morse and Feshbach (in 1953). The method proposed by Morse and Feshbach (in 1953) is realized by applying multi-dimensional integrals in space, which is impractical for large-scale 2D and 3D elastic reverse time migration because it requires O(N 2 ) computational complexity to calculate multi-dimensional integrals, where N is the total number of grid points. Zhang and McMechan (2010) proposed to directly apply Helmholtz decomposition to the coupled wave field in the wavenumber domain to generate the decomposed vector wave field. In addition, Helmholtz decomposition has also been extended to anisotropic media.
[0004] Most of the current wave field separation methods based on Helmholtz decomposition require solving a vector Poisson equation (e.g., Zhu, 2017). Zhu (2017) uses the discrete sine transform DST to solve equation (3). When discretizing using the finite difference method, Dirichlet boundary conditions are adopted without analyzing the boundary. The problems of the method are as follows: When separating the wave field, a vector Poisson equation needs to be solved, and the computational cost is very high, especially when solving three-dimensional problems. When solving the auxiliary wave field w, the discrete sine transform is used in the article, which further increases the cost. In addition, when discretizing, Dirichlet boundary conditions are considered, and the actual boundary conditions of w are not considered, so artifacts will appear in the finally separated P-wave field boundary and S-wave field boundary. Summary of the Invention
[0005] In view of the above problems, the purpose of the present invention is to provide an elastic wave field separation method, device and medium based on Helmholtz decomposition, which significantly improves the computational efficiency of the existing elastic wave field separation method based on Helmholtz decomposition, and no artifacts will appear in the finally separated P-wave field boundary and S-wave field boundary.
[0006] To achieve the above purpose, the present invention adopts the following technical solutions:
[0007] In the first aspect, the present invention provides an elastic wave field separation method based on Helmholtz decomposition, including the steps of:
[0008] Establish a functional relationship between the scalar function f and the wave field as shown in Equation (1):
[0009]
[0010] where u and u s are the input wave field and the S-wave vector field respectively, f is the scalar function, represents the gradient operator;
[0011] Take the divergence of both sides of Equation (1) to obtain a scalar Poisson equation, as shown in Equation (2):
[0012]
[0013] where u is the input wave field, f is the scalar function, represents the Laplace operator, represents the divergence operator;
[0014] Smoothly extend the divergence of the input wave field to obtain a smooth extension of the wave field divergence;
[0015] Utilize the connection between the Laplace operator in the spatial domain and the wave number domain, and combine the smooth extension of the wave field divergence to process the boundary to solve the scalar Poisson equation to obtain the scalar function f;
[0016] Perform elastic wavefield separation calculation according to the wavefield separation formula solved by the scalar function f.
[0017] Furthermore, the wavefield separation formula is Equation (3):
[0018]
[0019] where u p and u s are the P-wave vector field and the S-wave vector field respectively, f is the scalar function, and ▽ represents the gradient operator.
[0020] Furthermore, the smooth extension of the divergence of the input wavefield includes:
[0021] Use the finite difference method to calculate approximate approximation;
[0022] According to the Calculate the smooth extension of the divergence of the input wavefield.
[0023] Furthermore, the smooth extension of the divergence of the input wavefield also includes:
[0024] Retain the absorbing boundary of the input wavefield and use the finite difference method to calculate approximate approximation;
[0025] According to the Calculate the smooth extension of the divergence of the input wavefield as Equation (4):
[0026]
[0027] where f c is the truncation function, and u e represents the wavefield after retaining the absorbing boundary.
[0028] Furthermore, the calculation formula of the scalar function f is:
[0029]
[0030] where v1 is a smooth extension of the divergence of the input wavefield of, U is the region where the wavefield is located, k = |k|, k is the wave number, and F and F -1 are the Fourier transform and the inverse Fourier transform respectively.
[0031] In a second aspect, the present invention also provides an elastic wavefield separation device based on Helmholtz decomposition, including:
[0032] The first processing unit establishes a functional relationship between the scalar function f and the wavefield as Equation (1):
[0033]
[0034] Among them, u and u s are the input wave field and the S-wave vector field respectively, and f is a scalar function, represents the gradient operator.
[0035] The second processing unit takes the divergence of both sides of Equation (1) to obtain the scalar Poisson equation, as shown in Equation (2):
[0036]
[0037] Among them, u is the input wave field, and f is a scalar function, represents the Laplace operator, represents the divergence operator.
[0038] The third processing unit obtains a smooth extension of the wave field divergence by performing a smooth extension on the wave field divergence;
[0039] The fourth processing unit uses the connection between the Laplace operator in the spatial domain and the wave number domain, and combines the smooth extension of the wave field divergence to process the boundary to solve the scalar Poisson equation to obtain the scalar function f;
[0040] The fifth processing unit performs elastic wave field separation calculation according to the wave field separation formula solved by the scalar function f.
[0041] 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 elastic wave field separation method based on Helmholtz decomposition when being executed by a processor.
[0042] 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, wherein the processor implements the elastic wave field separation method based on Helmholtz decomposition when executing the computer program.
[0043] Due to the above technical solutions adopted by the present invention, it has the following advantages:
[0044] In wave field separation, a wave field separation method based on the scalar Poisson equation is proposed, which will significantly reduce the calculation cost compared with the method based on the vector Poisson equation;
[0045] By using the connection between the Laplace operator in the spatial domain and the wave number domain, we propose a smooth extension technique combined with a truncation function to handle general boundary problems, and finally no artifacts will appear in the separated P-wave field boundary and S-wave field boundary. Description of the Drawings
[0046] By reading the following detailed description of the preferred embodiments, various other advantages and benefits will become clear to those of ordinary skill in the art. The accompanying 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 denote the same components.
[0047] In the accompanying drawings:
[0048] Figure 1 is a flowchart of the elastic wave field separation method based on Helmholtz decomposition provided by the present invention;
[0049] Figure 2 are the vertical and horizontal components of displacement in the homogeneous model;
[0050] Figure 3 are the P-wave and S-wave separated without considering the boundary conditions during the decomposition process;
[0051] Figure 4 is to use the elastic wave field separation method based on Helmholtz decomposition proposed by the present invention for Figure 2 the P-wave and S-wave separated from the wave field therein;
[0052] Figure 5 are the vertical and horizontal components of displacement in the inhomogeneous model (Marmousi model);
[0053] Figure 6 Using the elastic wave field separation method based on Helmholtz decomposition proposed by the present invention for Figure 5 the P-wave and S-wave separated from the wave field therein;
[0054] Figure 7 is the P-wave and S-wave separated from the wave field in Figure 5 using the elastic wave field separation method of Helmholtz decomposition proposed by Zhu (2017). Detailed Embodiments
[0055] 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.
[0056] The present invention provides an elastic wave field separation method based on Helmholtz decomposition. In the wave field separation, by using the relationships among gradient, divergence, curl, and exterior derivative operations, a wave field separation method based on the scalar Poisson equation is proposed. Compared with the method based on the vector Poisson equation, the computational cost will be significantly reduced. The method also includes using the connection between the Laplace operator in the spatial domain and the wavenumber domain. We propose a smooth extension technique combined with a truncation function to handle general boundary problems, and no artifacts will appear in the separated P-wave field boundary and S-wave field boundary.
[0057] As Figure 1 described above, an embodiment of the present invention provides an elastic wave field separation method based on Helmholtz decomposition, including the steps:
[0058] S1: Establish the functional relationship between the scalar function f and the wave field as shown in Equation (1):
[0059]
[0060] The step S1 is based on the theorem: Assume that U is a contractible region. If the wave field u - u s satisfies in the region U , then there exists a function f such that
[0061]
[0062] The proof process of the theorem is prior art and will not be elaborated here.
[0063] S2. Take the divergence of both sides of Equation (1) to obtain the scalar Poisson equation, as shown in Equation (2):
[0064]
[0065] S3. Obtain the smooth extension of the wave field divergence through smooth extension of the wave field divergence;
[0066] S4. Utilize the connection between the Laplace operator in the spatial domain and the wavenumber domain, and combine the smooth extension of the wave field divergence to handle the boundary and solve the scalar Poisson equation to obtain the scalar function f;
[0067] S5. Perform elastic wave field separation calculation according to the wave field separation formula solved from the scalar function f.
[0068] Furthermore, the wave field separation formula is Equation (3):
[0069]
[0070] Furthermore, the smooth extension of the input wave field divergence includes:
[0071] Calculate using the finite difference method approximate approximation of;
[0072] According to the said calculate the smooth extension of the divergence of the input wave field.
[0073] Preferably, the smooth extension of the divergence of the input wave field further includes:
[0074] Retain the absorption boundary of the input wave field and use the finite difference method to calculate approximate approximation of;
[0075] According to the said calculate the smooth extension of the divergence of the input wave field as in Equation (4):
[0076]
[0077] where f c is a truncation function, and u e represents the wave field after retaining the absorption boundary.
[0078] The smooth extension of the divergence of the input wave field in the present invention is realized by retaining the absorption boundary and then combining with the truncation function. This process can also be realized by other methods, such as directly applying the truncation function to the input wave field near the boundary in the original region.
[0079] For the solution of the scalar Poisson equation, utilize the connection between the Laplace operator in the spatial domain and the wave number domain, and combine with the smooth extension technology of the truncation function to handle general boundary problems, and calculate the scalar function f:
[0080]
[0081] where v1 is a smooth extension of the divergence of the input wave field of, U is the region where the wave field is located, k = |k|, k is the wave, and F and F -1 are the Fourier transform and the inverse Fourier transform respectively.
[0082] Formula (5) means that after the smooth extension of the divergence of the input wave field, we can accurately solve the scalar Poisson equation (2) by means of the Fourier transform and the inverse Fourier transform. This solves the problem of how to solve the problem Poisson equation under general boundary conditions.
[0083] After solving the function f, the longitudinal wave and the transverse wave can be directly obtained using Equation (3).
[0084] Specific application examples of the said method are as follows:
[0085] Example 1
[0086] In Example 1, in the homogeneous medium model, it shows that when the parameters are V p = 3 km / s, V s = 2 km / s and ρ = 1.5 g / cm³, the P-wave and S-wave decompositions in the homogeneous model are presented. The source wavelet is a 15 Hz Ricker wavelet. The source-receiver distance is 1250 m and the depth is 1250 m. The vertical and horizontal component wavefields are as Figure 2 shown.
[0087] Based on the scalar Poisson equation (2), we use the proposed wavefield decomposition formula (3) to decompose the longitudinal and transverse wave displacements. In this example, we take the following functions as the truncation function to extend the divergence of the input wavefield:
[0088]
[0089] where i = 0, 1, 2,..., N, and N represents the number of extension layers. Other attenuation functions can also be used as the truncation function. Figure 3 and Figure 4 respectively give the decomposed wavefields without smooth extension and with smooth extension.
[0090] From Figure 3 , it can be seen that there are some artifacts around the boundary, which is due to the fact that the boundary conditions are not considered in the decomposition process. Figure 4 The P-waves and S-waves decomposed by using the proposed fast Poisson solver through the smooth extension technique in
[0091] Example 2
[0092] uses the heterogeneous medium (Marmousi) model. The source wavelet is a 10 Hz Ricker wavelet, the source-receiver distance is 6250 m, and the depth is 1750 m. The vertical and horizontal component wavefields are as Figure 5 shown. We use the wavefield decomposition formula (3) to decompose the longitudinal and transverse wave displacements based on the Poisson equation (2).
[0093] The P-wave fields and S-wave fields after decomposition without smooth extension and with smooth extension are respectively as Figure 6 and Figure 7 shown. In this experiment, the truncation function is as shown in formula (6).
[0094] The vertical and horizontal components of the displacement in the heterogeneous model (Marmousi model), Figure 6 are the P-waves and S-waves obtained by separating the wavefield in Figure 5 using the new method proposed in the present invention, Figure 7 and Figure 5The P-wave and S-wave obtained by separating the wave field therein.
[0095] Similar to the case of homogeneous medium, from Figure 7 we can see that there are some artifacts near the boundary of the decomposed waves, which are indicated by black arrows. However, it can be seen that in Figure 6 by using the smooth extension technique provided by the present invention, Figure 6 the artifacts around the boundary are effectively eliminated.
[0096] It should be noted that the smooth extension of the divergence of the input wave field in the present invention is achieved by retaining the absorption boundary and combining with a truncation function. This process can also be achieved by other methods, such as directly applying the truncation function to the input wave field near the boundary in the original region.
[0097] The choice of the truncation function is not unique, and other functions with decay properties can also be used as the truncation function.
[0098] The solution of the scalar Poisson equation is achieved by combining Fourier transform and inverse Fourier transform. The solution of this equation can also be achieved by other methods, such as discrete sine transform.
[0099] In addition, the present invention also provides an elastic wave field separation device based on Helmholtz decomposition, including:
[0100] The first processing unit establishes a functional relationship between the scalar function f and the wave field as shown in Equation (1):
[0101]
[0102] The second processing unit simultaneously takes the divergence of both sides of Equation (1) to obtain the scalar Poisson equation, as shown in Equation (2):
[0103]
[0104] The third processing unit obtains a smooth extension of the wave field divergence by performing a smooth extension on the wave field divergence;
[0105] The fourth processing unit utilizes the connection between the Laplace operator in the spatial domain and the wave number domain, and combines the smooth extension of the wave field divergence to solve the scalar Poisson equation at the boundary to obtain the scalar function f;
[0106] The fifth processing unit performs elastic wave field separation calculation according to the wave field separation formula solved by the scalar function f.
[0107] The present invention also provides a computer-readable storage medium storing computer instructions, and the computer instructions are used to implement the elastic wave field separation method based on Helmholtz decomposition when executed by a processor.
[0108] 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. When the processor executes the computer program, the elastic wave field separation method based on Helmholtz decomposition as described above is implemented.
[0109] In the wave field separation of the present invention, by using the relationships among gradient, divergence, curl, and exterior derivative operations, a wave field separation method based on the scalar Poisson equation is proposed. Compared with the method based on the vector Poisson equation, the computational cost will be significantly reduced. By using the connection between the Laplace operator in the spatial domain and the wavenumber domain, we propose a smooth extension technique combined with a truncation function to handle general boundary problems.
[0110] 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 described in the foregoing embodiments, or perform equivalent replacements for 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. An elastic wave field separation method based on Helmholtz decomposition, characterized in that Including the steps: Create a scalar function f The functional relationship between the function and the wave field is as shown in Equation (1): （1） wherein, and are the input wave field and the S-wave vector field respectively, f is a scalar function, denotes the gradient operator; Taking the divergence of both sides of Equation (1) to obtain the scalar Poisson equation as shown in Equation (2): （2） Among them, is the input wave field, f is a scalar function, represents the Laplace operator, represents the divergence operator; Obtaining a smooth extension of the wavefield divergence by performing a smooth extension on the divergence of the input wavefield; Using the connection between the Laplace operator in the spatial domain and the wavenumber domain, and combining the smooth extension of the divergence of the wave field to solve the boundary of the scalar Poisson equation to obtain the scalar function f ; According to the scalar function f perform elastic wavefield separation calculation according to the wavefield separation formula solved The smooth extension of the divergence of the input wavefield includes: Calculate using the finite difference method for the approximate approximation; According to the said calculate the smooth extension of the divergence of the input wave field; The smooth extension of the divergence of the input wavefield further includes: Retain the absorbing boundary of the input wave field and use the finite difference method to calculate approximate approximation; According to the above-mentioned The smooth extension for calculating the divergence of the input wave field is as shown in Equation (4): （4） Among them, is a truncation function, represents the wave field after retaining the absorbing boundary, is the divergence of the input wave field which is a smooth extension.
2. The elastic wave field separation method based on Helmholtz decomposition according to claim 1, wherein The wavefield separation formula is Equation (3): ， （3） Among them, and are the P-wave vector field and the S-wave vector field respectively, f is a scalar function, represents the gradient operator.
3. The elastic wave field separation method based on Helmholtz decomposition according to claim 1, wherein The scalar function f has the following calculation formula: （5） Among them, is the region where the wave field is located, , is the wave number, and are the Fourier transform and the inverse Fourier transform, respectively.
4. An elastic wave field separation device based on Helmholtz decomposition, characterized in that, Including: The first processing unit establishes a scalar function f and the functional relationship between the wave field is as shown in Equation (1): （1） Among them, and are the input wave field and the S-wave vector field respectively, f is a scalar function, represents the gradient operator; A second processing unit that takes the divergence of both sides of Equation (1) to obtain the scalar Poisson equation as shown in Equation (2): （2） Among them, is the input wave field, f is a scalar function, represents the Laplace operator, represents the divergence operator; A third processing unit that obtains a smooth extension of the wavefield divergence by performing a smooth extension on the divergence of the input wavefield; The fourth processing unit uses the connection between the Laplacian operator in the spatial domain and the wavenumber domain, and combines the smooth extension of the wave field divergence to process the boundary to solve the scalar Poisson equation to obtain the scalar function f ; The fifth processing unit performs elastic wavefield separation calculation according to the wavefield separation formula solved by the scalar function f ; The smooth extension of the divergence of the input wavefield includes: Calculate using the finite difference method for approximate approximation; According to the above-mentioned calculate the smooth extension of the divergence of the input wave field; The smooth extension of the divergence of the input wavefield further includes: Retain the absorbing boundary of the input wave field and use the finite difference method to calculate approximate approximation; According to the said The smooth extension for calculating the divergence of the input wave field is as shown in Equation (4): （4） Among them, is a truncation function, represents the wave field after retaining the absorbing boundary, is the divergence of the input wave field which is a smooth extension.
5. A computer-readable storage medium, characterized in that, Stored with computer instructions, the computer instructions are used to implement the elastic wavefield separation method based on Helmholtz decomposition as described in any one of Claims 1 to 3 when being executed by a processor.
6. A computer device, characterized in that, Including 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 elastic wavefield separation method based on Helmholtz decomposition as described in any one of Claims 1 to 3.