CN103412328A - Wave number field amplitude preservation wave field separation method based on staggered mesh finite difference algorithm - Google Patents

Abstract

The invention relates to a wave number field amplitude preservation wave field separation method based on a staggered mesh finite difference algorithm. The method includes the steps that the staggered mesh finite difference algorithm is used for conducting elastic wave field numerical simulation, the Fourier transform is used for transforming a wave field snapshoot from a spatial domain to a wave number field, an interpolation operator of the wave number field is used for interpolating a velocity component to the same grid node, a normalization wave number is used for conducting amplitude preservation wave field separation, and the Fourier inversion is used for transforming a separated result to the spatial domain to obtain amplitude preservation P waves and S waves. According to the wave number field amplitude preservation wave field separation method based on the staggered mesh finite difference algorithm, the staggered mesh finite difference algorithm is used so that high-precision seismic wave field numerical simulation can be achieved; the interpolation operator of the wave number field has high interpolation precision, and thus the parameter value of the same grid node can be well estimated; the normalization wave number is used so that amplitude preservation wave field separation can be achieved, and the wave fields of the amplitude preservation P waves and the S waves are obtained; due to the fact that wave field separation is conducted in the wave number field, and relative to the Fourier transform and the Fourier inversion, the calculated amount can not be increased obviously in wave number field interpolation.

Description

Wavenumber domain based on the staggered-mesh finite-difference algorithm is protected amplitude wave field separation method

Technical field

The invention belongs to the exploration geophysics field, particularly, relate to a kind of wavenumber domain based on the staggered-mesh finite-difference algorithm and protect amplitude wave field separation method.

Background technology

Wave field separation is one of important technology in field of seismic exploration, all plays a very important role at aspects such as Forward Problem of Vsp research, multi-component seismic data processing and explanations.The wave field separation method mainly is divided into two large classes at present: the first kind is, on the inspection surface of earth's surface, many component recordings are carried out to wave field separation, and Equations of The Second Kind is, in the elastic wave field communication process, the wave field snapshot is carried out to wave field separation.Industry member is general direct using the horizontal component of many component recordings as the S ripple, and vertical component does not utilize ripe compressional wave treatment scheme to process as the P wavelength-division.This kind way is to be based upon on the hypothesis basis on seismic event vertical outgoing earth's surface after the weathering zone of earth's surface.But the in fact impossible fully vertical earth's surface outgoing of ray, so horizontal component and vertical component all comprise P ripple and S ripple.Therefore this simple approximate certain noise and the illusion of will inevitably causing disturbed.Another kind of method of separating more accurately many component recordings is to carry out polarization filtering.At first estimate P ripple and S wave polarization information, then on their polarization direction, carry out respectively projection, thereby obtain P ripple and the S ripple record separated.Therefore but because the ground wave field information is incomplete, lack the information of depth direction, actual, must adopt certain hypothesis while asking for polarization direction.Because actual underground medium is that flexible, seismic wave field is vector wave, therefore in the wave field communication process, the wave field snapshot is carried out to the feature that wave field separation can keep elastic wave and vector wave better.Usually utilize the Helmholtz decomposition to ask for divergence to the wave field snapshot and obtain with curl P ripple and the S ripple separated, but the P ripple obtained and and the S ripple at aspects such as numerical value, dimensions, change has occurred all with respect to original input wave field, make the wave field separation result protect the width Shortcomings.Aspect wave field numerical, finite-difference algorithm is a kind of important tool.With respect to the conventional difference algorithm, the staggered-mesh finite-difference algorithm has higher precision under the prerequisite of identical calculations amount due to it, therefore is widely used.But the wave field parameter-definition of this algorithm is on different grid nodes, and wave field can't directly separate.

Summary of the invention

In order to overcome, utilize staggered-mesh method of finite difference elastic wave field numerical modeling result to carry out the deficiency of the parameter-definition of wave field separation at different grid nodes, in order to overcome, usually utilize Helmholtz to decompose to obtain the poor deficiency of separation wave field guarantor's width simultaneously, the present invention proposes a kind of wavenumber domain based on the staggered-mesh finite-difference algorithm and protect amplitude wave field separation method, by at first utilizing wavenumber domain interpolation operator estimation same mesh node parameter value, thereby and then utilize the normalization wave number to carry out P ripple and S ripple that wave field separation obtains guarantor's width.

For achieving the above object, technical scheme of the present invention is as follows:

Wavenumber domain based on the staggered-mesh finite-difference algorithm is protected amplitude wave field separation method, it is characterized in that, comprises the following steps:

Step 1: utilize the staggered-mesh finite-difference algorithm to carry out elastic wave field numerical modeling

Step 2: utilize Fourier transform that the wave field snapshot is transformed from a spatial domain to wavenumber domain

Step 3: utilize the wavenumber domain interpolation operator that speed component is interpolated into to identical grid node

Step 4: utilize the normalization wave number to carry out the Bao Fubochang separation

Step 5: utilize Fourier inversion that separating resulting is transformed to spatial domain and obtain P ripple and the S ripple of protecting width.

With respect to prior art, beneficial effect of the present invention is as follows: utilize the staggered-mesh finite-difference algorithm can realize the seismic wave field numerical simulation of degree of precision; The interpolation operator of wavenumber domain has very high interpolation precision, therefore can well estimate the parameter value of same mesh node; Utilize the normalization wave number can realize protecting the wave field separation of width, obtain P ripple and the S wave field of protecting width; Because wave field separation is carried out at wavenumber domain, therefore with respect to positive and negative Fourier transform, the wavenumber domain interpolation can't significantly increase calculated amount.

The accompanying drawing explanation

Fig. 1 is based on the process flow diagram of the wavenumber domain guarantor amplitude wave field separation method of staggered-mesh finite-difference algorithm;

Fig. 2 is Particle Vibration Velocity component distribution plan in staggered-mesh;

Fig. 3 (a) is a certain moment x of depression model component wave field snapshot;

Fig. 3 (b) is a certain moment z of depression model component wave field snapshot;

Fig. 4 (a) utilizes to protect amplitude wave field separation method gained P ripple result;

Fig. 4 (b) utilizes to protect amplitude wave field separation method gained S ripple result.

Embodiment

As shown in Figure 1, protect amplitude wave field separation method based on the wavenumber domain of staggered-mesh finite-difference algorithm, comprise the following steps:

Step 1: utilize the staggered-mesh finite-difference algorithm to carry out elastic wave field numerical modeling

Utilize the staggered-mesh finite difference scheme to carry out elastic wave one-order velocity-stress equation discrete, obtain the elastic wave propagation operator; Load given source wavelet, utilize the elastic wave propagation operator to carry out elastic wave field numerical modeling.Concrete grammar is as follows:

Utilize elastodynamic three fundamental equations: the geometric equation of describing the constitutive equation of stress-strain relation, the moving equilibrium differential equation of describing the stress displacement relation and description displacement strain stress relation obtains elastic wave one-order velocity-stress equation:

$\frac{\∂V(x,z,t)}{\∂t}=(A^x\frac{\∂}{\∂x}+A^z\frac{\∂}{\∂z})V(x,z,t),---\left(1\right)$

Wherein,

$V=\left[\begin{array}{c}{v}_x\\ v_z\\ {\mathrm{\τ}}_{\mathrm{xx}}\\ {\mathrm{\τ}}_{\mathrm{zz}}\\ {\mathrm{\τ}}_{\mathrm{xz}}\end{array}\right],A^x=\left[\begin{array}{ccccc}0& 0& \frac{1}{\mathrm{\ρ}}& 0& 0\\ 0& 0& 0& 0& \frac{1}{\mathrm{\ρ}}\\ C_{11}& 0& 0& 0& 0\\ C_{13}& 0& 0& 0& 0\\ 0& C_{44}& 0& 0& 0\end{array}\right],A^z=\left[\begin{array}{ccccc}0& 0& 0& 0& \frac{1}{\mathrm{\ρ}}\\ 0& 0& 0& \frac{1}{\mathrm{\ρ}}& 0\\ 0& C_{13}& 0& 0& 0\\ 0& C_{33}& 0& 0& 0\\ C_{44}& 0& 0& 0& 0\end{array}\right],---\left(2\right)$

V _x, v _zBe respectively horizontal component and the vertical component of Particle Vibration Velocity, τ _IjFor the components of stress, C _IjFor elastic constant, ρ is Media density.

Utilize the staggered-mesh finite difference scheme to carry out elastic wave one-order velocity-stress equation (1) discrete, obtain the elastic wave propagation operator:

$V_{i,j}^{n+1}=V_{i,j}^n+\mathrm{\Δt}{A}^x{L}_x\left(V_{\mathrm{ai},j}^{n+\frac{1}{2}}\right)+\mathrm{\Δ}{\mathrm{tA}}^z{L}_z\left(V_{\mathrm{bi},j}^{n+\frac{1}{2}}\right),---\left(3\right)$

Wherein,

$\left\{\begin{array}{c}{V}_{\mathrm{ai},j}^n={[v_{\mathrm{xi}+\frac{1}{2},j}^n,v_{\mathrm{zi},j+\frac{1}{2}}^n,{\mathrm{\τ}}_{\mathrm{xxi},j}^n,{\mathrm{\τ}}_{\mathrm{zzi},j}^n,{\mathrm{\τ}}_{\mathrm{xzi}+\frac{1}{2},j+\frac{1}{2}}^n]}^T\\ V_{\mathrm{bi},j}^n={[v_{\mathrm{xi},j+\frac{1}{2}}^n,v_{\mathrm{zi}+\frac{1}{2},j}^n,{\mathrm{\τ}}_{\mathrm{xxi},j}^n,{\mathrm{\τ}}_{\mathrm{zzi}+\frac{1}{2},j+\frac{1}{2}}^n,{\mathrm{\τ}}_{\mathrm{xzi},j}^n]}^T\end{array}\right.,---\left(4\right)$

L _x, L _zFor 2L rank, space finite difference operator.Because but the 2L rank precision difference approximate representation of staggered-mesh first order derivative is:

$\mathrm{\Δx}{f}^{\left(1\right)}\left(x_0\right)=\underset{m=1}{\overset{L}{\mathrm{\Σ}}}{a}_m[f(x_0+\frac{2m-1}{2}\mathrm{\Δx})-f(x_0-\frac{2m-1}{2}\mathrm{\Δx})],---\left(5\right)$

Wherein, Δ x is mesh spacing, a _mFor high-order staggered-mesh difference coefficient.Therefore, L _x, L _zBe expressed as follows:

$\left\{\begin{array}{c}{L}_x\left(V_{i,j}^n\right)=\frac{1}{\mathrm{\Δx}}\underset{m=1}{\overset{L}{\mathrm{\Σ}}}{a}_m(V_{i+\frac{2m-1}{2},j}^n-V_{i-\frac{2m-1}{2},j}^n)\\ L_z\left(V_{i,j}^n\right)=\frac{1}{\mathrm{\Δz}}\underset{m=1}{\overset{L}{\mathrm{\Σ}}}{a}_m(V_{i,j+\frac{2m-1}{2}}^n-V_{i,j-\frac{2m-1}{2}}^n)\end{array}\right..---\left(6\right)$

Wherein, a _mBy following equation, determine:

Load given source wavelet, utilize elastic wave propagation operator (3), can realize elastic wave field numerical modeling, build underground elasticity vector wave field.

Step 2: utilize Fourier transform that the wave field snapshot is transformed from a spatial domain to wavenumber domain

Utilize the Fourier transform wave field snapshot that each time step of step 1 gained is corresponding to transform from a spatial domain to wavenumber domain.Concrete grammar is as follows:

2D Fourier direct transform is

$\tilde{V}(k_x,k_z)=\underset{-\∞}{\overset{+\∞}{\∫}}\underset{-\∞}{\overset{+\∞}{\∫}}v(x,z)e^{-i2\mathrm{\π}(k_{x}x+k_{z}z)}\mathrm{dxdz},---\left(8\right)$

The 2D Fourier inversion is

$v(x,z)=\underset{-\∞}{\overset{+\∞}{\∫}}\underset{-\∞}{\overset{+\∞}{\∫}}(k_x,k_z)e^{i2\mathrm{\π}(k_{x}x+k_{z}z)}d{k}_{x}d{k}_z,---\left(9\right)$

Wherein, v is spatial domain Particle Vibration Velocity parameter,

For its corresponding wavenumber domain result, x, z are respectively the coordinate of spatial domain horizontal direction and vertical direction, k _x, k _zBe respectively the wave number of wavenumber domain horizontal direction and vertical direction.

Utilize 2D Fourier direct transform (8) the wave field snapshot that each time step is corresponding to transform from a spatial domain to wavenumber domain.

Step 3: utilize the wavenumber domain interpolation operator that speed component is interpolated into to identical grid node

The wavenumber domain result that each time step of step 2 gained is corresponding is multiplied by corresponding interpolation operator, and the wavenumber domain result is interpolated on identical grid node.Concrete grammar is as follows:

As shown in Figure 2, due to v in staggered-mesh _xAnd v _zComponent is defined on different grid nodes, the v at grid node O place _xComponent is with the 2D Fourier inversion (10) of horizontal direction interpolation operator, estimating

$v_x(x+\mathrm{\Δx}/2,z)=\stackrel{+\∞}{\underset{-\∞}{\∫}}\stackrel{+\∞}{\underset{-\∞}{\∫}}{\tilde{V}}_x(k_x,k_z)e^{i2\mathrm{\π}[k_x(x+\mathrm{\Δx}/2)+k_{z}z]}d{k}_{x}d{k}_z,---\left(10\right)$

The v at grid node O place _zComponent is with the 2D Fourier inversion (11) of vertical direction interpolation operator, estimating

$v_z(x+\mathrm{\Δx}/2,z)=\stackrel{+\∞}{\underset{-\∞}{\∫}}\stackrel{+\∞}{\underset{-\∞}{\∫}}{\tilde{V}}_z(k_x,k_z)e^{i2\mathrm{\π}[k_{x}x+k_z(z-\mathrm{\Δz}/2)]}d{k}_{x}d{k}_z,---\left(11\right)$

Wherein,

With

Respectively with Fig. 2 in v _x(x, z) and v _z(x+ Δ x/2, z+ Δ z/2) correspondence.

At wavenumber domain, will

Be multiplied by interpolation operator

And will

Be multiplied by interpolation operator They can be interpolated into to same position.

Step 4: utilize the normalization wave number to carry out the Bao Fubochang separation

Utilize the normalization wave number to realize that at wavenumber domain Bao Fubochang separates the wavenumber domain wave field component after step 3 gained interpolation.Concrete grammar is as follows:

To spatial domain wave field V, ask divergence to obtain the P ripple

$P=\frac{{\∂V}_x}{\∂x}+\frac{\∂V_z}{\∂z},---\left(12\right)$

To spatial domain wave field V, ask curl to obtain the S ripple

$S=(\frac{{\∂V}_x}{\∂z}-\frac{{\∂V}_z}{\∂x})a_y,---\left(13\right)$

Wherein, V _x, V _zBe respectively horizontal component and the vertical component of wave field V, a _yFor the unit vector of y direction, now the S ripple can be considered scalar wave.To ask divergence gained P wave field (12) to carry out 2D Fourier direct transform to the wave field snapshot and obtain corresponding wavenumber domain result:

$\tilde{P}={\mathrm{ik}}_x{\tilde{V}}_x+{\mathrm{ik}}_z{\tilde{V}}_z,---\left(14\right)$

To ask curl gained S wave field (13) to carry out 2D Fourier direct transform to the wave field snapshot and obtain corresponding wavenumber domain result:

$\tilde{S}=\left(i{k}_z{\tilde{V}}_x-i{k}_x\widetilde{V_z}\right)a_y,---\left(15\right)$

Wherein,

With

Be respectively the x wavenumber domain result corresponding with the z component of spatial domain wave field, k _xWith k _zBe respectively the wave number of x direction and z direction.

In spatial domain, ask partial derivative to be equivalent at wavenumber domain and be multiplied by corresponding wave number, this can make the wave field after separation all change at aspects such as amplitude and units with respect to original wave field.In order to proofread and correct, ask the impact of partial derivative on wave field strength and unit, will With

Bring P ripple Bao Fubochang into and separate the P ripple result that formula (16) obtains guarantor's width

$\tilde{P}=i\frac{k_x}{k}{\tilde{V}}_x+i\frac{k_z}{k}{\tilde{V}}_z,---\left(16\right)$

Will

With

Bring S ripple Bao Fubochang into and separate the S ripple result that formula (17) obtains guarantor's width

$\tilde{S}=i(\frac{k_z}{k}{\tilde{V}}_x-\frac{k_x}{k}{\tilde{V}}_z)a_y,---\left(17\right)$

Wherein, Absolute value for wave number.

Step 5: utilize Fourier inversion that separating resulting is transformed to spatial domain and obtain P ripple and the S ripple of protecting width

Utilize 2D Fourier inversion (9) to change to spatial domain the result that step 4 gained wavenumber domain separates, thereby obtain P ripple and the S wave field of protecting width.

Accompanying drawing 4(a) and 4(b) be respectively through P ripple corresponding to the step-length sometime of above-mentioned steps gained and S ripple result, wave field has obtained good separation and this result and accompanying drawing 3(a) and 3(b) in the input snapshot have identical magnitude and unit, wave field realize to be protected width and is separated.

Claims (6)

1. the wavenumber domain based on the staggered-mesh finite-difference algorithm is protected amplitude wave field separation method, it is characterized in that, comprises the following steps:

Step 1: utilize the staggered-mesh finite-difference algorithm to carry out elastic wave field numerical modeling

Step 2: utilize Fourier transform that the wave field snapshot is transformed from a spatial domain to wavenumber domain

Step 3: utilize the wavenumber domain interpolation operator that speed component is interpolated into to identical grid node

Step 4: utilize the normalization wave number to carry out the Bao Fubochang separation

Step 5: utilize Fourier inversion that separating resulting is transformed to spatial domain and obtain P ripple and the S ripple of protecting width.

2. the wavenumber domain based on the staggered-mesh finite-difference algorithm according to claim 1 is protected amplitude wave field separation method, it is characterized in that, step 1 is: utilize the staggered-mesh finite difference scheme to carry out elastic wave one-order velocity-stress equation discrete, obtain the elastic wave propagation operator; Load given source wavelet, utilize the elastic wave propagation operator to carry out elastic wave field numerical modeling; Concrete grammar is as follows:

Utilize elastodynamic three fundamental equations: the geometric equation of describing the constitutive equation of stress-strain relation, the moving equilibrium differential equation of describing the stress displacement relation and description displacement strain stress relation obtains elastic wave one-order velocity-stress equation:

$\frac{\∂V(x,z,t)}{\∂t}=(A^x\frac{\∂}{\∂x}+A^z\frac{\∂}{\∂z})V(x,z,t)$

Wherein,

$V=\left[\begin{array}{c}{v}_x\\ v_z\\ {\mathrm{\τ}}_{\mathrm{xx}}\\ {\mathrm{\τ}}_{\mathrm{zz}}\\ {\mathrm{\τ}}_{\mathrm{xz}}\end{array}\right],A^x=\left[\begin{array}{ccccc}0& 0& \frac{1}{\mathrm{\ρ}}& 0& 0\\ 0& 0& 0& 0& \frac{1}{\mathrm{\ρ}}\\ C_{11}& 0& 0& 0& 0\\ C_{13}& 0& 0& 0& 0\\ 0& C_{44}& 0& 0& 0\end{array}\right],A^z=\left[\begin{array}{ccccc}0& 0& 0& 0& \frac{1}{\mathrm{\ρ}}\\ 0& 0& 0& \frac{1}{\mathrm{\ρ}}& 0\\ 0& C_{13}& 0& 0& 0\\ 0& C_{33}& 0& 0& 0\\ C_{44}& 0& 0& 0& 0\end{array}\right]$

V _x, v _zBe respectively horizontal component and the vertical component of Particle Vibration Velocity, τ _IjFor the components of stress, C _IjFor elastic constant, ρ is Media density;

Utilize the staggered-mesh finite difference scheme to carry out elastic wave one-order velocity-stress equation discrete, obtain the elastic wave propagation operator:

$V_{i,j}^{n+1}=V_{i,j}^n+\mathrm{\Δt}{A}^x{L}_x\left(V_{\mathrm{ai},j}^{n+\frac{1}{2}}\right)+\mathrm{\Δ}{\mathrm{tA}}^z{L}_z\left(V_{\mathrm{bi},j}^{n+\frac{1}{2}}\right)$

Wherein,

$\left\{\begin{array}{c}{V}_{\mathrm{ai},j}^n={[v_{\mathrm{xi}+\frac{1}{2},j}^n,v_{\mathrm{zi},j+\frac{1}{2}}^n,{\mathrm{\τ}}_{\mathrm{xxi},j}^n,{\mathrm{\τ}}_{\mathrm{zzi},j}^n,{\mathrm{\τ}}_{\mathrm{xzi}+\frac{1}{2},j+\frac{1}{2}}^n]}^T\\ V_{\mathrm{bi},j}^n={[v_{\mathrm{xi},j+\frac{1}{2}}^n,v_{\mathrm{zi}+\frac{1}{2},j}^n,{\mathrm{\τ}}_{\mathrm{xxi},j}^n,{\mathrm{\τ}}_{\mathrm{zzi}+\frac{1}{2},j+\frac{1}{2}}^n,{\mathrm{\τ}}_{\mathrm{xzi},j}^n]}^T\end{array}\right.$

L _x, L _zFor 2L rank, space finite difference operator; Because but the 2L rank precision difference approximate representation of staggered-mesh first order derivative is:

$\mathrm{\Δx}{f}^{\left(1\right)}\left(x_0\right)=\underset{m=1}{\overset{L}{\mathrm{\Σ}}}{a}_m[f(x_0+\frac{2m-1}{2}\mathrm{\Δx})-f(x_0-\frac{2m-1}{2}\mathrm{\Δx})]$

Wherein, Δ x is mesh spacing, a _mFor high-order staggered-mesh difference coefficient; Therefore, L _x, L _zBe expressed as follows:

$\left\{\begin{array}{c}{L}_x\left(V_{i,j}^n\right)=\frac{1}{\mathrm{\Δx}}\underset{m=1}{\overset{L}{\mathrm{\Σ}}}{a}_m(V_{i+\frac{2m-1}{2},j}^n-V_{i-\frac{2m-1}{2},j}^n)\\ L_z\left(V_{i,j}^n\right)=\frac{1}{\mathrm{\Δz}}\underset{m=1}{\overset{L}{\mathrm{\Σ}}}{a}_m(V_{i,j+\frac{2m-1}{2}}^n-V_{i,j-\frac{2m-1}{2}}^n)\end{array}\right.$

Wherein, a _mBy following equation, determine:

Load given source wavelet, utilize the elastic wave propagation operator, can realize elastic wave field numerical modeling, build underground elasticity vector wave field.

3. according to the described wavenumber domain based on the staggered-mesh finite-difference algorithm of claim 1-2, protect amplitude wave field separation method, it is characterized in that, step 2 is to utilize the Fourier transform wave field snapshot that each time step of step 1 gained is corresponding to transform from a spatial domain to wavenumber domain; Concrete grammar is as follows:

2D Fourier direct transform is

$\tilde{V}(k_x,k_z)=\underset{-\∞}{\overset{+\∞}{\∫}}\underset{-\∞}{\overset{+\∞}{\∫}}v(x,z)e^{-i2\mathrm{\π}(k_{x}x+k_{z}z)}\mathrm{dxdz}$

The 2D Fourier inversion is

$v(x,z)=\stackrel{+\∞}{\underset{-\∞}{\∫}}\stackrel{+\∞}{\underset{-\∞}{\∫}}\tilde{V}(k_x,k_z)e^{i2\mathrm{\π}(k_{x}x+k_{z}z)}d{k}_{x}d{k}_z$

Wherein, v is spatial domain Particle Vibration Velocity parameter,

For its corresponding wavenumber domain result, x, z are respectively the coordinate of spatial domain horizontal direction and vertical direction, k _x, k _zBe respectively the wave number of wavenumber domain horizontal direction and vertical direction;

Utilize the 2D Fourier direct transform wave field snapshot that each time step is corresponding to transform from a spatial domain to wavenumber domain.

4. according to the described wavenumber domain based on the staggered-mesh finite-difference algorithm of claim 1-3, protect amplitude wave field separation method, it is characterized in that, step 3 is multiplied by corresponding interpolation operator for the wavenumber domain result that each time step of step 2 gained is corresponding, and the wavenumber domain result is interpolated on identical grid node; Concrete grammar is as follows:

Due to v in staggered-mesh _xAnd v _zComponent is defined on different grid nodes, the v at grid node O place _xComponent is with the 2D Fourier inversion of horizontal direction interpolation operator, estimating:

$v_x(x+\mathrm{\Δx}/2,z)=\stackrel{+\∞}{\underset{-\∞}{\∫}}\stackrel{+\∞}{\underset{-\∞}{\∫}}{\tilde{V}}_x(k_x,k_z)e^{i2\mathrm{\π}[k_x(x+\mathrm{\Δx}/2)+k_{z}z]}d{k}_{x}d{k}_z$

The v at grid node O place _zComponent is with the 2D Fourier inversion of vertical direction interpolation operator, estimating:

$v_z(x+\mathrm{\Δx}/2,z)=\stackrel{+\∞}{\underset{-\∞}{\∫}}\stackrel{+\∞}{\underset{-\∞}{\∫}}{\tilde{V}}_z(k_x,k_z)e^{i2\mathrm{\π}[k_{x}x+k_z(z-\mathrm{\Δz}/2)]}d{k}_{x}d{k}_z$

Wherein,

With

Respectively with v _x(x, z) and v _z(x+ Δ x/2, z+ Δ z/2) correspondence;

At wavenumber domain, will

Be multiplied by interpolation operator

And will

Be multiplied by interpolation operator

They can be interpolated into to same position.

5. according to the described wavenumber domain based on the staggered-mesh finite-difference algorithm of claim 1-4, protect amplitude wave field separation method, it is characterized in that, step 4 utilizes the normalization wave number to realize that at wavenumber domain Bao Fubochang separates the wavenumber domain wave field component after step 3 gained interpolation; Concrete grammar is as follows:

V asks divergence to obtain the P ripple to spatial domain wave field snapshot

$P=\frac{\∂V_x}{\∂x}+\frac{\∂V_z}{\∂z}$

V asks curl to obtain the S ripple to spatial domain wave field snapshot

$S=(\frac{\∂V_x}{\∂z}-\frac{\∂V_z}{\∂x})a_y$

Wherein, V _x, V _zBe respectively horizontal component and the vertical component of wave field V, a _yFor the unit vector of y direction, now the S ripple can be considered scalar wave;

To ask divergence gained P wave field to carry out 2D Fourier direct transform to the wave field snapshot and obtain corresponding wavenumber domain result

$\tilde{P}=i{k}_x{\tilde{V}}_x+i{k}_z{\tilde{V}}_z$

To ask curl gained S wave field to carry out 2D Fourier direct transform to the wave field snapshot and obtain corresponding wavenumber domain result

$\tilde{S}=(i{k}_z{\tilde{V}}_x-i{k}_x{\tilde{V}}_z)a_y$

Wherein,

With

Be respectively the x wavenumber domain result corresponding with the z component of spatial domain wave field, k _xWith k _zBe respectively the wave number of x direction and z direction;

In spatial domain, ask partial derivative to be equivalent at wavenumber domain and be multiplied by corresponding wave number, this can make the wave field after separation all change at aspects such as amplitude and units with respect to original wave field; In order to proofread and correct, ask the impact of partial derivative on wave field strength and unit, will

With

Bring P ripple Bao Fubochang into and separate formula

$\tilde{P}=i\frac{k_x}{k}{\tilde{V}}_x+i\frac{k_z}{k}{\tilde{V}}_z$

Must protect the P ripple result of width; Will

With

Bring S ripple Bao Fubochang into and separate formula

$\tilde{S}=i(\frac{k_z}{k}{\tilde{V}}_x-\frac{k_x}{k}{\tilde{V}}_z)a_y$

Must protect the S ripple result of width; Wherein,

Absolute value for wave number.

6. according to the described wavenumber domain based on the staggered-mesh finite-difference algorithm of claim 1-5, protect amplitude wave field separation method, it is characterized in that, step 5 utilizes the 2D Fourier inversion to change to spatial domain for the result that step 4 gained wavenumber domain is separated, thereby obtains P ripple and the S wave field of protecting width.

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