CN104597488B - Optimum design method of finite difference template of non-equiangular long-grid wave equation - Google Patents
Optimum design method of finite difference template of non-equiangular long-grid wave equation Download PDFInfo
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Abstract
The invention discloses an optimum design method of a finite difference template of a non-equiangular long-grid wave equation. The method comprises the following steps: according to a time sampling interval and a spatial sampling interval, mesh subdivision is executed to the simulating region of an actual geologic model; according to the maximum appointed permissible error and the wave number range, the corresponding operator lengths of different acoustic velocities are calculated; based on the finite difference method of the least square optimization time space domain and the corresponding operator length, the optimized finite difference coefficient of the grid is acquired; the acquired optimized finite difference coefficient is brought into the wave equation with a difference scheme to perform forward modeling of the wave equation. According to the method, the finite difference method of the time space domain only applied to square and cube grids is expanded into the rectangular or rectangular parallelepiped grids, thus a special simulation precision requirement and a requirement for saving a calculated amount are met in actual production.
Description
Technical field
The present invention relates to seismic forward modeling numerical simulation technology field, more particularly to a kind of non-equilateral wavelength grid wave equation
Finite difference optimum formwork design method.
Background technology
Seismic forward modeling numerical simulation technology be complex underground structure group (including isotropic medium, anisotropic medium,
Biot multiphases anisotropic medium, random hole medium etc.) it is known in the case of, make ripple in this Jie using numerical computation method
Propagated in matter, through multiple transmission, reflection, the scattering of subsurface geological structure, the mistake received by the wave detector of earth's surface or underground
Journey.The seismic response of underground complex geological structure is simulated using accurate wave equation numerical solution, is that Study of Seismic ripple is passed
Broadcast the mathematics thing that many aspects such as the explanation of mechanism, the special treatment method of seismic data and bad ground provide more science
Reason foundation.In recent years, Wave Equation Numerical method is widely used in reverse-time migration and full waveform inversion.
Wave equation forward modeling has various methods, and more typical has：Finite difference method, pseudo- spectrometry, FInite Element, boundary element
Method, spectral element method etc..Wherein finite difference method is because its amount of calculation is small, computational efficiency is high, be adapted to more complicated rate pattern
And be widely used.Finite difference calculus can be divided into according to different standards：Explicit finite difference and implicit finite difference；Rule
Grid finite difference, staggering mesh finite-difference and rotationally staggered grid finite difference.In finite difference calculus, difference coefficient can be with
Tried to achieve by Taylor series expansion or optimal method, finite difference based on Taylor series expansion is corresponded to respectively and with most
Finite difference based on optimization.In conventional finite calculus of finite differences, difference coefficient is obtained by the dispersion relation of minimization spatial domain
Arrive.In recent years, a kind of time-space domain finite difference calculus is occurred in that, the method is closed by the frequency dispersion of minimization time-domain and spatial domain
System asks for difference coefficient, with simulation precision and more preferable stability higher.
Spatial sampling interval in current time-space domain finite difference method requirement all directions is equal, that is, needs mould
Type subdivision is square or square volume mesh.And in actual production, in order to meet specific required precision or in order to save meter
Calculation amount, we are usually needed model facetization, and for rectangle or rectangular parallelepiped grid, (the grid spacing of typically depth direction is different from
Horizontal direction).Suitable for square and the time-space domain finite difference method of square volume mesh, it is impossible to meet specific required precision
Or amount of calculation can not be saved.
The content of the invention
A kind of non-equilateral wavelength grid wave equation finite difference optimum formwork design method is the embodiment of the invention provides, will
It is only applicable to during the time-space domain finite difference method of square and square volume mesh expands to rectangle or rectangular parallelepiped grid, meets real
The demand of specific simulation precision requirement and saving amount of calculation in the production of border, the method includes：
Mesh generation is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval；
According to the given limits of error and wave-number range, corresponding operator length is obtained to different SVELs；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, obtain the excellent of grid
Change finite difference coefficient；
The optimization finite difference coefficient that will be obtained substitutes into the wave equation of difference scheme, carries out Wave equation forward modeling；
It is described that difference net is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval
Lattice subdivision, including：The simulated domain of actual geological model is split into rectangular parallelepiped grid；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, obtain the excellent of grid
Change finite difference coefficient, including：Obtain the optimization finite difference coefficient of rectangular parallelepiped grid；
The optimization finite difference coefficient for obtaining rectangular parallelepiped grid, calculates according to equation below：
Wherein,
Wherein, b is wave number, and M is operator length, amBe the finite difference coefficient after optimization, θ be plane wave propagation direction with
The angle of horizontal plane, θ ∈ [0, π]；φ is the azimuth of plane wave propagation, φ ∈ [0,2 π]；V is SVEL, τ
It is time sampling interval, h is x, y direction sampling interval；C=Δs z/h, c are parametric variable, and Δ z is the z directions sampling interval；β=
Kh, k are parametric variable, and β is wave-number range, β ∈ [0, b]；M is parametric variable, and m is integer, m ∈ [1, M]；N is parametric variable,
N is integer, n ∈ [1, M].
In one embodiment, the corresponding worst error of the optimization finite difference coefficient of the rectangular parallelepiped grid meets as follows
Constraints：
ξ1max<η；
Wherein, ξ1maxIt is the corresponding worst error of optimization finite difference coefficient of rectangular parallelepiped grid；η is maximum allowable mistake
Difference.
In one embodiment, the corresponding worst error ξ of optimization finite difference coefficient of the rectangular parallelepiped grid1maxBy such as
Lower formula is calculated：
Wherein,
In one embodiment, the optimization finite difference coefficient that will be obtained substitutes into the wave equation of difference scheme, enters
Row Wave equation forward modeling, including：The optimization finite difference coefficient of the rectangular parallelepiped grid that will be obtained substitutes into the three of difference scheme
In dimension Acoustic Wave-equation, three-dimensional acoustic wave Wave equation forward modeling is carried out；The three-dimensional acoustic wave wave equation of the difference scheme
For：
Wherein, P is acoustic pressure.
The embodiment of the invention provides a kind of non-equilateral wavelength grid wave equation finite difference optimum formwork design method, bag
Include：
Mesh generation is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval；
According to the given limits of error and wave-number range, corresponding operator length is obtained to different SVELs；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, obtain the excellent of grid
Change finite difference coefficient；
The optimization finite difference coefficient that will be obtained substitutes into the wave equation of difference scheme, carries out Wave equation forward modeling；
It is described that difference net is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval
Lattice subdivision, including：The simulated domain of actual geological model is split into rectangular mesh；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, obtain the excellent of grid
Change finite difference coefficient, including：Obtain the optimization finite difference coefficient of rectangular mesh；
The optimization finite difference coefficient for obtaining rectangular mesh, calculates according to equation below：
Wherein,
Wherein, b is wave number, and M is operator length, amBe the finite difference coefficient after optimization, θ be plane wave propagation direction with
The angle of horizontal plane, θ ∈ [0, π]；V is SVEL, and τ is time sampling interval, and h is the x directions sampling interval；C=
Δ z/h, c are parametric variable, and Δ z is the z directions sampling interval；β=kh, k are parametric variable, and β is wave-number range, β ∈ [0, b]；m
It is parametric variable, m is integer, m ∈ [1, M]；N is parametric variable, and n is integer, n ∈ [1, M].
In one embodiment, the corresponding worst error of the optimization finite difference coefficient of the rectangular mesh meets as follows about
Beam condition：
ξ2max<η；
Wherein, ξ2maxIt is the corresponding worst error of optimization finite difference coefficient of rectangular mesh；η is the limits of error.
In one embodiment, the corresponding worst error ξ of optimization finite difference coefficient of the rectangular mesh2maxBy as follows
Formula is calculated：
Wherein,
In one embodiment, in the wave equation of the optimization finite difference coefficient substitution difference scheme that will be obtained,
Wave equation forward modeling is carried out, is also included：The optimization finite difference coefficient of the rectangular mesh that will be obtained substitutes into difference scheme
In two-dimentional Acoustic Wave-equation, two-dimentional Acoustic Wave-equation forward simulation is carried out；The two-dimentional sound wave fluctuation side of the difference scheme
Cheng Wei：
Wherein, P is acoustic pressure.
In embodiments of the present invention, it is proposed that a kind of non-equilateral wavelength grid wave equation finite difference optimum formwork design side
Method, the method will be only applicable to square and the time-space domain finite difference method of square volume mesh extend to rectangle or cuboid
In grid, the demand of specific simulation precision requirement and saving amount of calculation in actual production is met.
Brief description of the drawings
Accompanying drawing described herein is used for providing a further understanding of the present invention, constitutes the part of the application, not
Constitute limitation of the invention.In the accompanying drawings：
Fig. 1 is the non-equilateral wavelength grid wave equation finite difference optimum formwork design side of one kind provided in an embodiment of the present invention
Method flow chart；
Fig. 2 is the error curve obtained by traditional finite difference method provided in an embodiment of the present invention with the direction of propagation
Changing Pattern figure；
Fig. 3 is provided in an embodiment of the present invention by non-equilateral wavelength grid wave equation finite difference optimum formwork design method
The error curve of acquisition with the direction of propagation Changing Pattern figure；
Fig. 4 is the error curve obtained by traditional finite difference method provided in an embodiment of the present invention with operator length
Changing Pattern figure；
Fig. 5 is provided in an embodiment of the present invention by non-equilateral wavelength grid wave equation finite difference optimum formwork design method
The error curve of acquisition with operator length Changing Pattern figure；
Fig. 6 is that provided in an embodiment of the present invention have by traditional finite difference method and non-equilateral wavelength grid wave equation
Wave field snapshot comparison diagram of the uniform sound wave medium that limit difference template Optimization Design is obtained respectively at the 1.0s moment；
Fig. 7 is the comparison of wave shape figure of dotted line position in Fig. 6；
Fig. 8 is that provided in an embodiment of the present invention have by traditional finite difference method and non-equilateral wavelength grid wave equation
Wave field snapshot comparison diagram of the Marmousi models that limit difference template Optimization Design is obtained respectively at the 1.0s moment；
Specific embodiment
It is right with reference to implementation method and accompanying drawing to make the object, technical solutions and advantages of the present invention become more apparent
The present invention is described in further details.Here, exemplary embodiment of the invention and its illustrating for explaining the present invention, but simultaneously
It is not as a limitation of the invention.
Inventor has found that the spatial sampling interval in current time-space domain finite difference method requirement all directions is equal,
It is square or square volume mesh exactly to need model facetization, but this kind of method can not meet specific required precision or
Amount of calculation can not be saved.If by model facetization be rectangle or rectangular parallelepiped grid, can meet specific required precision or
Save amount of calculation.Based on this, the present invention proposes a kind of non-equilateral wavelength grid wave equation finite difference optimum formwork design method.
Fig. 1 is the non-equilateral wavelength grid wave equation finite difference optimum formwork design side of one kind provided in an embodiment of the present invention
Method flow chart, as shown in figure 1, the method includes：
Step 101：Net is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval
Lattice subdivision；
Step 102：According to the given limits of error and wave-number range, corresponding calculation is obtained to different SVELs
Sub- length；
Step 103：Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, obtain
The optimization finite difference coefficient of grid；
Step 104：The optimization finite difference coefficient that will be obtained substitutes into the wave equation of difference scheme, is carrying out wave equation just
Drill simulation.
, it is necessary to be rectangular mesh or length by simulated domain (or zoning) subdivision of actual geological model during specific implementation
Cube grid (typically the grid spacing of depth direction is different from horizontal direction).
Three-dimensional acoustic wave wave equation is as follows：
In formula, P represents acoustic pressure, and V represents SVEL.
In case of z directions grid spacing is different from x, y direction, the optimization finite difference of specific rectangular parallelepiped grid
Form is as follows：
Wherein, τ is time sampling interval；H is x, y direction sampling interval；C=Δs z/h, c are parametric variable, and Δ z is z side
To the sampling interval；amIt is the finite difference coefficient after optimization, M represents operator length；M is parametric variable, and m is integer, m ∈ [1,
M]。
In order to simplify symbol, we willIt is abbreviated asAccording to Plane wave theory, Wo Menling：
Wherein,
Wherein, i, l, j, n are parametric variable, and n is integer, n ∈ [1, M]；K is parametric variable；θ ∈ [0, π] are plane wave
The direction of propagation and the angle of horizontal plane, φ ∈ [0,2 π] are the azimuth of plane wave propagation.Equation (3), (4) are substituted into equation
(2), and appropriate abbreviation, can obtain：
Wherein,Further according to constraints；
Can obtain：
Wherein, β=kh, β are wave-number range, β ∈ [0, b].Optimization finite-difference algorithm seeks to obtain a set of difference system
Number causes that equation (7) two ends error is minimum.According to Least Square Theory, this problem can be converted into solution equation below group：
Wherein,
By solving system of linear equations (8), we can obtain the optimization difference coefficient of rectangular parallelepiped grid.Again by difference system
Number is updated in equation (2), it is possible to carry out the solution of three-dimensional acoustic wave wave equation.
In order to ensure the validity of the inventive method, it is necessary to add following constraint bar during optimization above
The corresponding worst error of optimization finite difference coefficient of part, i.e. rectangular parallelepiped grid meets following constraints：
ξ1max<η (10)
We pass through the corresponding worst error of following formula calculation optimization difference coefficient：
Wherein,
Wherein, η is the limits of error.
ξ1maxIt is only relevant with b and M.When M is fixed, ξ1maxIt is decided by b, ξ1maxBecome big as b increases.Therefore, if initially
B be unsatisfactory for equation (10), we should be by being gradually reduced b until ξ1max<η obtains optimal b.In addition, when b is fixed, ξ1maxOnly
It is relevant with M, ξ1maxDiminish as M increases.So, if initial M is unsatisfactory for equation (10), we should be by gradually increasing M
Until ξ1max<η obtains optimal M.
For two-dimensional rectangle grid, in case of z directions grid spacing is different from x directions, specifically optimize limited
Difference scheme is as follows：
In formula, P represents acoustic pressure, and V represents SVEL；Wherein, τ is time sampling interval；H is the x directions sampling interval；c
=Δ z/h, c are parametric variable, and Δ z is the z directions sampling interval；amIt is the finite difference coefficient after optimization, M represents that operator is long
Degree；M is parametric variable, and m is integer, m ∈ [1, M].
We can obtain the difference coefficient of the optimization of rectangular mesh by solving following linear equation：
Wherein,
By solving system of linear equations (15), we can obtain the optimization difference coefficient of rectangular mesh.Again by difference system
Number is updated in equation (13), it is possible to carry out the solution of two-dimentional Acoustic Wave-equation.
Likewise, in order to ensure the validity of the inventive method, it is necessary to be added during optimization above following
The corresponding worst error of optimization finite difference coefficient of constraints, i.e. rectangular mesh meets following constraints：
ξ2max<η (16)
We pass through the corresponding worst error of following formula calculation optimization difference coefficient：
Wherein,
Wherein, η is the limits of error.
Below advantage of the invention is illustrated by taking a uniform sound wave medium as an example.Velocity of longitudinal wave is 1500m/s, and the time adopts
At intervals of 1ms, the x directions sampling interval is 10m to sample, and the z directions sampling interval is 5m, operator length M ∈ [2,10].
Regulation b=2.74 and M=8, using the inventive method, (i.e. non-equilateral wavelength grid wave equation finite difference template is excellent
Change method for designing) and traditional finite difference method obtain Changing Pattern of the error curve with the direction of propagation, such as Fig. 2 and Tu respectively
Shown in 3.Regulation η=10-2.5≈ 0.003, error curve is obtained with calculation using the inventive method and traditional finite difference method
The Changing Pattern of sub- length, respectively as shown in Figures 4 and 5.From Fig. 2 to Fig. 5, with the time-space domain finite difference side based on Taylor
Method (traditional finite difference method) is compared, and the inventive method has broader effective band and smaller numerical solidification.Regulation η
=10-2.5≈ 0.003, the wave field snapshot pair of uniform sound wave medium is obtained using traditional finite difference method and the inventive method
Than scheming, as shown in Figure 6.Wherein, the left side figure in Fig. 6 is the ripple that uniform sound wave medium is obtained using traditional finite difference method
Field snapshot, the right figure is the wave field snapshot of the uniform sound wave medium obtained using the inventive method.Fig. 7 is dotted line place in Fig. 6
The comparison of wave shape figure of position.From Fig. 6 and Fig. 7, when operator length is identical, the inventive method simulation precision is higher.
In order to further illustrate effectiveness of the invention, we are just drilled with Marmousi models.Time sampling interval
It is 1ms, the x directions sampling interval is 12m, and the z directions sampling interval is 9m.The operator length M=9 of traditional finite difference calculus.When
Spatial domain finite difference selects the operator of different length, M ∈ [2,9] according to different speed using the algorithm for becoming operator length.Figure
8 is the Marmousi models obtained respectively by traditional finite difference method and the inventive method provided in an embodiment of the present invention
In the wave field snapshot comparison diagram at 1.0s moment, upper graph is that the Marmousi models obtained by traditional finite difference method are existed
The wave field snapshot at 1.0s moment, lower edge graph is the ripple of the Marmousi models that are obtained respectively by the inventive method at the 1.0s moment
Field snapshot；As shown in Figure 8, the inventive method is applied to the forward simulation of complex dielectrics, and its simulation effect is substantially better than biography
The finite difference calculus of system.
In sum, the inventive method has advantages below：1. the numerical solidification of medium-high frequency section is reduced.2. with time-space domain
More meet reality based on dispersion relation, simulation precision is higher.3. it is applied to rectangle and rectangular parallelepiped grid, actual life can be met
The demand of specific required precision and saving amount of calculation in product.
The preferred embodiments of the present invention are the foregoing is only, is not intended to limit the invention, for the skill of this area
For art personnel, the embodiment of the present invention can have various modifications and variations.It is all within the spirit and principles in the present invention, made
Any modification, equivalent substitution and improvements etc., should be included within the scope of the present invention.
Claims (8)
1. a kind of non-equilateral wavelength grid wave equation finite difference optimum formwork design method, it is characterised in that including：
Mesh generation is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval；
According to the given limits of error and wave-number range, corresponding operator length is obtained to different SVELs；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, the optimization for obtaining grid have
Limit difference coefficient；
The optimization finite difference coefficient that will be obtained substitutes into the wave equation of difference scheme, carries out Wave equation forward modeling；
It is described difference gridding is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval to cut open
Point, including：The simulated domain of actual geological model is split into rectangular parallelepiped grid；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, the optimization for obtaining grid have
Limit difference coefficient, including：Obtain the optimization finite difference coefficient of rectangular parallelepiped grid；
The optimization finite difference coefficient for obtaining rectangular parallelepiped grid, calculates according to equation below：
Wherein,
Wherein, b is wave number, and M is operator length, amIt is the finite difference coefficient after optimization, θ is plane wave propagation direction and level
The angle in face, θ ∈ [0, π]；φ is the azimuth of plane wave propagation, φ ∈ [0,2 π]；V is SVEL, when τ is
Between the sampling interval, h be x, y direction sampling interval；C=Δs z/h, c are parametric variable, and Δ z is the z directions sampling interval；β=kh, k
It is parametric variable, β is wave-number range, β ∈ [0, b]；M is parametric variable, and m is integer, m ∈ [1, M]；N is parametric variable, and n is
Integer, n ∈ [1, M].
2. the method for claim 1, it is characterised in that the optimization finite difference coefficient of the rectangular parallelepiped grid is corresponding
Worst error meets following constraints：
ξ1max<η；
Wherein, ξ1maxIt is the corresponding worst error of optimization finite difference coefficient of rectangular parallelepiped grid；η is the limits of error.
3. method as claimed in claim 2, it is characterised in that the optimization finite difference coefficient of the rectangular parallelepiped grid is corresponding
Worst error ξ1maxIt is calculated as follows：
Wherein,
4. method as claimed in claim 3, it is characterised in that the optimization finite difference coefficient that will be obtained substitutes into difference lattice
The wave equation of formula, carries out Wave equation forward modeling, including：The optimization finite difference coefficient generation of the rectangular parallelepiped grid that will be obtained
Enter in the three-dimensional acoustic wave wave equation of difference scheme, carry out three-dimensional acoustic wave Wave equation forward modeling；The three of the difference scheme
Tieing up Acoustic Wave-equation is：
Wherein, P is acoustic pressure.
5. a kind of non-equilateral wavelength grid wave equation finite difference optimum formwork design method, it is characterised in that including：
Mesh generation is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval；
According to the given limits of error and wave-number range, corresponding operator length is obtained to different SVELs；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, the optimization for obtaining grid have
Limit difference coefficient；
The optimization finite difference coefficient that will be obtained substitutes into the wave equation of difference scheme, carries out Wave equation forward modeling；
It is described difference gridding is carried out to the simulated domain of actual geological model according to time sampling interval and spatial sampling interval to cut open
Point, including：The simulated domain of actual geological model is split into rectangular mesh；
Time-space domain finite difference calculus and described corresponding operator length based on Least-squares minimization, the optimization for obtaining grid have
Limit difference coefficient, including：Obtain the optimization finite difference coefficient of rectangular mesh；
The optimization finite difference coefficient for obtaining rectangular mesh, calculates according to equation below：
Wherein,
Wherein, b is wave number, and M is operator length, amIt is the finite difference coefficient after optimization, θ is plane wave propagation direction and level
The angle in face, θ ∈ [0, π]；V is SVEL, and τ is time sampling interval, and h is the x directions sampling interval；C=Δs z/
H, c are parametric variable, and Δ z is the z directions sampling interval；β=kh, k are parametric variable, and β is wave-number range, β ∈ [0, b]；M is ginseng
Number variable, m is integer, m ∈ [1, M]；N is parametric variable, and n is integer, n ∈ [1, M].
6. method as claimed in claim 5, it is characterised in that the optimization finite difference coefficient of the rectangular mesh is corresponding most
Big error meets following constraints：
ξ2max<η；
Wherein, ξ2maxIt is the corresponding worst error of optimization finite difference coefficient of rectangular mesh；η is the limits of error.
7. method as claimed in claim 6, it is characterised in that the optimization finite difference coefficient of the rectangular mesh is corresponding most
Big error ξ2maxIt is calculated as follows：
Wherein,
8. method as claimed in claim 7, it is characterised in that the optimization finite difference coefficient that will be obtained substitutes into difference lattice
In the wave equation of formula, Wave equation forward modeling is carried out, also included：The optimization finite difference coefficient of the rectangular mesh that will be obtained
Substitute into the two-dimentional Acoustic Wave-equation of difference scheme, carry out two-dimentional Acoustic Wave-equation forward simulation；The difference scheme
Two-dimentional Acoustic Wave-equation is：
Wherein, P is acoustic pressure.
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| CN107976710B (en) * | 2017-11-17 | 2019-05-28 | 河海大学 | A kind of implicit time-space domain finite difference numerical simulation method of linear optimization based on ACOUSTIC WAVE EQUATION |
| CN108279437A (en) * | 2018-01-17 | 2018-07-13 | 中国石油大学(华东) | Variable density ACOUSTIC WAVE EQUATION time higher order accuracy staggering mesh finite-difference method |
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