MEDIA PARAMETER-MODIFIED METHOD FOR REALIZING AN ADAPTIVE EXPRESSION OF AN ARBITRARY DISCONTINUOUS SURFACE

Abstract

A media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface, comprising the following steps: importing an initial forward model, importing anisotropic parameters; and setting a space step and a time step according to the initial forward model parameters; and then starting a stepped discretization of a free surface of the initial forward model; and using a corrected constitutive relationship to correct a first level parameter of the initial forward model; and bringing the corrected constitutive relationship into a displacement stress equation, and the influence of the free surface can be introduced in the case of the anisotropic media after series of operation. The present disclosure can make an accurate numerical simulation of a wave field near the discontinuous surface, and the accurate numerical simulation will contribute to the extraction and analysis of information from the seismic data.

Claims

1. A media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface, comprising the following steps: S1: importing an initial forward model, and the initial forward model includes a velocity and a density of vertical and horizontal waves, setting a seismic wavelet type and a main frequency parameter, and setting a space step and a time step according to initial forward model parameters; S2: selecting the space step and the time step according to the initial forward model parameters; and setting a differential order-number and a seismic wavelet; S3: starting a stepped discretization of a free surface of the initial forward model when a surface of the initial forward model fluctuates, and turning an irregular undulating ground into a regular stepped grid; S4: using a media Parameter-modified method to correct a constitutive relationship and a density of non-horizontal points on the free surface; S5: using a corrected constitutive relationship to replace an original constitutive relationship on the free surface, and an influence of the free surface can be introduced without correcting an original finite difference method code; and S6: for generalized anisotropic media, including orthorhombic anisotropic media and triclinic anisotropic media, bringing the corrected constitutive relationship into a finite difference method code to introduce the influence of the free surface.

2. The media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface according to claim 1, wherein: the free surface includes a fluid free surface and a solid free surface.

3. The media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface according to claim 1, wherein: a discontinuous surface includes a fluid-solid discontinuous surface and a fluid-vacuum discontinuous surface.

4. The media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface according to claim 1, wherein: for the anisotropic media, anisotropic parameters need to be imported.

5. The media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface according to claim 1, wherein: the anisotropic media is suitable for a solid free surface part.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0020] FIG. 1 is a flow chart of the media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface in an embodiment of the present disclosure;

[0021] FIG. 2 shows a schematic diagram of free surface discretization in the embodiment of the present disclosure;

[0022] FIG. 3 shows a schematic diagram of the positional relationship between solid and air in the embodiment of the present disclosure; graph a) shows that on the left side of the free surface is air (use VL for short); graph b) shows that on the back side of the free surface is air (use VB for short); Similarly, VR: vertical boundary grid cell with air to the right; H: horizontal boundary grid cell; VF: vertical boundary grid cell with air to the front.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0023] In order to make the objects, technical solutions, and advantages of the present disclosure clear, the present disclosure will be further described in detail below with reference to the embodiments and drawings. The exemplary embodiments of the present disclosure and the description thereof are only used to explain the present disclosure, not as a limitation of the present disclosure.

[0024] A media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface, comprising the following steps:

[0025] S1: importing an initial forward model, and the initial forward model includes a velocity and a density of vertical and horizontal waves, setting a seismic wavelet type and a main frequency parameter, and setting a space step and a time step according to initial forward model parameters;

[0026] S2: selecting the space step and the time step according to the initial forward model parameters; and setting a differential order-number and a seismic wavelet;

[0027] S3: starting a stepped discretization of a free surface of the initial forward model when a surface of the initial forward model fluctuates, and turning an irregular undulating ground into a regular stepped grid;

[0028] S4: using the media Parameter-modified method to correct a constitutive relationship and a density of non-horizontal points on the free surface;

[0029] S5: using a corrected constitutive relationship to replace an original constitutive relationship on the free surface, and an influence of the free surface can be introduced without correcting an original finite difference method code;

[0030] S6: for generalized anisotropic media, including orthorhombic anisotropic media and triclinic anisotropic media, bringing the corrected constitutive relationship into a finite difference method code can introduce the influence of the free surface.

[0031] Further, a discontinuous surface includes a fluid-solid discontinuous surface and a fluid-vacuum discontinuous surface; and the anisotropic media is suitable for solid free surface part.

[0032] Further, for the solid-fluid discontinuous surface, the constitutive relationship corresponding to a solid media 1 is assumed to be τ.sup.−M1=E.sup.M1(λ.sub.1, μ.sub.1)ε.sup.−M1, the constitutive relationship corresponding to a solid media 2 is τ.sup.−M2=E.sup.M2(λ.sub.2, μ.sub.2)ε.sup.−M2. According to an average media thought, two connected media can be equivalent to an equivalent average media, and the constitutive relationship of the equivalent average media is τ.sup.−A=E.sup.A(λ.sub.1,μ.sub.1,λ.sub.2, μ.sub.2)ε.sup.−A. Similarly, for the solid-fluid discontinuous surface, the constitutive relationship corresponding to a fluid media 1 is supposed to be τ.sup.−M1=E.sup.M1(λ.sub.1,μ.sub.1)ε.sup.−M1, the constitutive relationship corresponding to a fluid media 2 is τ.sup.−M2=E.sup.M2(λ.sub.2,0)ε.sup.−M2. According to the average media thought, two connected media can be equivalent to the equivalent average media, and the constitutive relationship of the equivalent average media is τ.sup.−A=E.sup.A(λ.sub.1,μ.sub.1,λ.sub.2,0)ε.sup.−A.

[0033] In the formula above, τ refers to a stress tensor matrix, M1 refers to a elastic media 1, M2 refers to a elastic media 2, A refers to the average media, E refers to a equivalent elastic coefficient matrix, λ.sub.1, λ.sub.2 and μ.sub.1, μ.sub.2a refer to a Lame constants, and ε refers to a matrix of strain.

[0034] Further, for the solid free surface such as the land surface, which are also called solid-vacuum discontinuous surface, the constitutive relationship of an upper vacuum media is assumed to be τ.sup.−M1=E.sup.M1(λ.sub.1,μ.sub.1)ε.sup.−M1, and according to λ.sub.1.fwdarw.0,μ.sub.1=0, the constitutive relationship of a sublayer solid media is τ.sup.−M2=E.sup.M2(λ.sub.2,0)ε.sup.−M2, the constitutive relationship of the equivalent average media is τ.sup.−A=E.sup.A(λ.sub.1,μ.sub.1,λ.sub.2,μ.sub.2)ε.sup.−A, and λ.sub.1.fwdarw.0,μ.sub.1=0.

[0035] Further, for a sublayer anisotropic media (Using VTI media as an example), which are also called solid-vacuum discontinuous surface, the constitutive relationship of the upper vacuum media is supposed to be τ.sup.−M1=E.sup.M1(c.sub.11.sup.−,c.sub.13.sup.−,c.sub.33.sup.−,c.sub.44.sup.−)ε.sup.−M1, and c.sub.33.sup.−.fwdarw.0,c.sub.44.sup.−=0. The constitutive relationship of the sublayer anisotropic media is τ.sup.−M2=E.sup.M2(c.sub.11.sup.+,c.sub.13.sup.+,c.sub.33.sup.+,c.sub.44.sup.+)ε.sup.−M2, and the constitutive relationship of equivalent average media is τ.sup.−A=E.sup.A(c.sub.11.sup.−,c.sub.13.sup.−,c.sub.33.sup.−,c.sub.44.sup.−,c.sub.11.sup.+,c.sub.13.sup.+,c.sub.33.sup.+,c.sub.44.sup.+), and c.sub.33.sup.−.fwdarw.0,c.sub.44.sup.−=0.

[0036] In the formula above, the elastic parameter of the anisotropic media is c.sub.11,c.sub.13,c.sub.33,c.sub.44.

[0037] In the formula above, τ refers to a stress tensor matrix, C refers to elastic coefficient, M1 refers to a elastic media 1, M2 refers to a elastic media 2, A refers to the average media, E refers to a equivalent elastic coefficient matrix: λ.sub.1, λ.sub.2 and μ.sub.1, μ.sub.2 refer to a Lame constants, and ε refers to a matrix of strain tensor.

[0038] In S1, for the anisotropic media, the anisotropic media c.sub.11,c.sub.13,c.sub.33,c.sub.44 need to be imported. Then setting parameters such as differential order-number, type of seismic source, and seismic source frequency, and selecting the space step and the time step according to the initial forward model parameters.

[0039] Further, for Orthorhombic anisotropic media or Triclinic anisotropic media, the constitutive relationship of the upper vacuum media is supposed to be τ.sup.−M1=E.sup.M1(c.sub.11.sup.−,c.sub.12.sup.−,c.sub.13.sup.−, c.sub.22.sup.−,c.sub.23.sup.−,c.sub.33.sup.−,c.sub.44.sup.−,c.sub.55.sup.−,c.sub.66.sup.−)ε.sup.−M1. The constitutive relationship of the sublayer anisotropic media is τ.sup.−M2=E.sup.M2(c.sub.11.sup.+,c.sub.12.sup.+,c.sub.13.sup.+,c.sub.22.sup.+,c.sub.23.sup.+,c.sub.33.sup.+,c.sub.44.sup.+,c.sub.55.sup.+,c.sub.66.sup.+)ε.sup.−M2, and the constitutive relationship of equivalent average media is τ.sup.−A=E.sup.A(c.sub.11.sup.−,c.sub.12.sup.−,c.sub.13.sup.−,c.sub.22.sup.−,c.sub.23.sup.−,c.sub.33.sup.−,c.sub.44.sup.−,c.sub.55.sup.−,c.sub.66.sup.−,c.sub.11.sup.+,c.sub.12.sup.+,c.sub.13.sup.+,c.sub.22.sup.+,c.sub.23.sup.+,c.sub.33.sup.+,c.sub.44.sup.+,c.sub.55.sup.+,c.sub.66.sup.+)ε.sup.−A.

[0040] Further, starting a stepped discretization of a free surface of the initial forward model when a surface of the initial forward model fluctuates, and the discretization is shown in FIG. 1. And as shown in FIG. 2, turning an irregular undulating ground into a regular stepped grid, OL, OR in FIG. 2 refer to an outer corner point of grid point, and IL, IR refer to an inner corner point of grid point. FIG. 2 displays the implementation of undulating topography, divides the grid points of the free surface, and modifies the constitutive relations of different grid points according to Table 1.

[0041] In S4, using the media Parameter-modified method to correct a constitutive relationship and a density of non-horizontal points on the free surface, the constitutive relationship can be seen in table 1. In table 1, VL shows that on the left side of the free surface is air, VR shows that on the right side of the free surface is air, VB shows that behind the free surface is air, and H shows that above the free surface is air. Refer to FIG. 3 for more details. In FIG. 3, A/F represents the air or the fluid and S represents an underground media.

TABLE-US-00001 TABLE 1 the constitutive relationship of the undulating topography at the free surface; Categories τ τ τ τ τ τ ρ.sub.x ρ.sub.y ρ.sub.z VL 0 βε  + αε αε  + βε 2με  με 2με   ρ.sub.x 0.5ρ.sub.y 0.5ρ.sub.z VF αε  + βε 0 βε  + αε 2με 2με  με 0.5ρ.sub.x   ρ.sub.y 0.5ρ.sub.z VR 0 βε  + αε αε  + βε 0  με 0 0 0.5ρ.sub.y 0.5ρ.sub.z VB αε  + βε 0 βε  + αε 0 0  με 0.5ρ.sub.x 0 0.5ρ.sub.z H αε  + βε βε  + αε 0  με 2με 2με 0.5ρ.sub.x 0.5ρ.sub.y   ρ.sub.z indicates data missing or illegible when filed

[0042] In table 1, τ.sub.ij refers to ijth component of the stress tensor, ε.sub.ij refers to ijth component of the strain tensor, λ an μ refer to the Lame constants; and ρ refers to density of the media; and α=(2μ(λ+μ)/(λ+2μ)), β=(μλ)/(λ+2μ).

[0043] Further, using the corrected constitutive relationship to modify the first-level parameters (the parameters at the free surface) of the initial forward model, and the influence of the free surface can be introduced without correcting the original finite difference method code. The constitutive relationship at the free surface is as follows:

[00001]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

[0044] u.sub.i refers to ith component of displacement, τ.sub.ij refers to ijth component of the stress tensor, ε.sub.ij refers to ijth component of the strain tensor, λ and μ refer to the Lame constants; and ρ refers to density of the media.

[0045] Further, for the anisotropic media like a VTI media, the corrected constitutive relationship of VTI anisotropic media is as follows:

[00002]$\left[\begin{array}{c}{\tau }_{\mathrm{xx}}\\ {\tau }_{\mathrm{yy}}\\ {\tau }_{\mathrm{zz}}\\ {\tau }_{\mathrm{xy}}\\ {\tau }_{\mathrm{yz}}\\ {\tau }_{\mathrm{zx}}\end{array}\right]={\left[\begin{array}{cc}\left(c_{11}{c}_{33}-c_{13}^2\right)/2c_{33}& \left(\left(c_{11}-2c_{66}\right)c_{33}-c_{13}^2\right)/2c_{33}\\ \left(\left(c_{11}-2c_{66}\right)c_{33}-c_{13}^2\right)/2c_{33}& \left(c_{11}{c}_{33}-c_{13}^2\right)/2c_{33}\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& c_{66}/2\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\mathrm{xx}}\\ {\epsilon }_{\mathrm{yy}}\\ {\epsilon }_{\mathrm{zz}}\\ 2{\epsilon }_{\mathrm{xy}}\\ 2{\epsilon }_{\mathrm{yz}}\\ 2{\epsilon }_{\mathrm{zx}}\end{array}\right]}_{z=0}$

[0046] Bringing the corrected constitutive relationship into a displacement stress equation:

[00003]$\{\begin{array}{c}{\frac{\rho }{2}\frac{{\partial }^2{u}_x}{\partial t^2}.Math.}_{z=0}=\frac{\partial {\tau }_{\mathrm{xx}}}{\partial x}+\frac{\partial {\tau }_{\mathrm{xy}}}{\partial y}+\frac{\partial {\tau }_{\mathrm{xz}}}{\partial z},\\ {\frac{\rho }{2}\frac{{\partial }^2{u}_y}{\partial t^2}.Math.}_{z=0}=\frac{\partial {\tau }_{\mathrm{yx}}}{\partial x}+\frac{\partial {\tau }_{\mathrm{yy}}}{\partial y}+\frac{\partial {\tau }_{\mathrm{yz}}}{\partial z},\\ {\frac{\rho }{2}\frac{{\partial }^2{u}_z}{\partial t^2}.Math.}_{z=0}=\frac{\partial {\tau }_{\mathrm{zx}}}{\partial x}+\frac{\partial {\tau }_{\mathrm{zy}}}{\partial y}+\frac{\partial {\tau }_{\mathrm{yz}}}{\partial z},\end{array}$$\{\begin{array}{c}{{\tau }_{\mathrm{xx}}.Math.}_{z=0}=\frac{\left(c_{11}{c}_{33}-c_{13}^2\right)}{2c_{33}}{\epsilon }_{\mathrm{xx}}+\frac{\left(\left(c_{11}-2c_{66}\right)c_{33}-c_{13}^2\right)}{2c_{33}}{\epsilon }_{\mathrm{yy}}\\ {{\tau }_{\mathrm{yy}}.Math.}_{z=0}=\frac{\left(\left(c_{11}-2c_{66}\right)c_{33}-c_{13}^2\right)}{2c_{33}}{\epsilon }_{\mathrm{xx}}+\frac{\left(c_{11}{c}_{33}-c_{13}^2\right)}{2c_{33}}{\epsilon }_{\mathrm{yy}}\\ {{\tau }_{\mathrm{zz}}.Math.}_{z=0}=0\\ {{{\tau }_{\mathrm{xy}}.Math.}_{z=0}={\tau }_{\mathrm{yx}}.Math.}_{z=0}=c_{66}{\epsilon }_{\mathrm{yx}}=c_{66}{\epsilon }_{\mathrm{xy}}\\ {{{\tau }_{\mathrm{yz}}.Math.}_{z=0}={\tau }_{\mathrm{zy}}.Math.}_{z=0}=0\\ {{{\tau }_{\mathrm{xz}}.Math.}_{z=0}={\tau }_{\mathrm{zx}}.Math.}_{z=0}=0\end{array}$

[0047] After the calculation of above formulas, the influence of the free surface in the case of the anisotropic media can be introduced.

[0048] The media Parameter-modified method for realizing an adaptive expression of an arbitrary discontinuous surface can be directly inherited and applied to the existing mainstream velocity-stress staggered grid finite fractional numerical program in the industry and academia, and the method can accurately fit the spatial position of the discontinuous surface because the method has no traditional grid offset error. The method can be achieved by simply correcting the media parameters of the grid points near the discontinuous surface, and the correction process only needs to be done once, the calculation is efficient and the operation is simple. The method can be applied to the high-order finite difference operator, achieving calculation accuracy equivalent to that based on the precise implementation of the weak solutions form of partial differential equations. The method is suitable for the discontinuous surface in the vicinity of the anisotropic media, and having a Poisson ration adaptive features. At the same time, this method is suitable to deal with various common anisotropic media free surface, such as VTI, HTI, TTI media, etc. Further, it can be extended to generalized anisotropic media, such as orthorhombic anisotropic media, triclinic anisotropic media, etc. The method is adaptive to Poisson's ratio for both isotropic and anisotropic media.

[0049] Although the preferred embodiments of the present disclosure have been described, but the skilled in the art can make additional changes and modifications to the embodiments once the skilled learn the basic creative concept. Therefore, claims are required to explain including preferred embodiments and all changes and modifications within the scope of the present disclosure.

[0050] Obviously, the skilled in the art can make various changes and modifications to the present disclosure without departing from the spirit and scope of the disclosure. If the modifications and changes of the present disclosure fall within the scope of the claims of the present disclosure and the claims' equivalent technologies, the present disclosure also intends to include the changes and modifications.