CN104597488A - 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
- Publication number
- CN104597488A CN104597488A CN201510029000.9A CN201510029000A CN104597488A CN 104597488 A CN104597488 A CN 104597488A CN 201510029000 A CN201510029000 A CN 201510029000A CN 104597488 A CN104597488 A CN 104597488A
- Authority
- CN
- China
- Prior art keywords
- mrow
- msubsup
- msup
- msub
- mfrac
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
- 238000000034 method Methods 0.000 title claims abstract description 87
- 238000013461 design Methods 0.000 title claims abstract description 14
- 238000005070 sampling Methods 0.000 claims abstract description 38
- 238000004088 simulation Methods 0.000 claims abstract description 26
- 238000005457 optimization Methods 0.000 claims abstract description 24
- 230000002123 temporal effect Effects 0.000 claims 1
- 238000004519 manufacturing process Methods 0.000 abstract description 5
- 238000004364 calculation method Methods 0.000 description 6
- 238000010586 diagram Methods 0.000 description 6
- 239000006185 dispersion Substances 0.000 description 5
- 230000008569 process Effects 0.000 description 3
- 238000004422 calculation algorithm Methods 0.000 description 2
- 230000004048 modification Effects 0.000 description 2
- 238000012986 modification Methods 0.000 description 2
- 230000005540 biological transmission Effects 0.000 description 1
- 230000008859 change Effects 0.000 description 1
- 230000003247 decreasing effect Effects 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 230000006872 improvement Effects 0.000 description 1
- 230000007246 mechanism Effects 0.000 description 1
- 230000005012 migration Effects 0.000 description 1
- 238000013508 migration Methods 0.000 description 1
- 238000003672 processing method Methods 0.000 description 1
- 230000001902 propagating effect Effects 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 230000004044 response Effects 0.000 description 1
- 230000003595 spectral effect Effects 0.000 description 1
Landscapes
- Geophysics And Detection Of Objects (AREA)
- Investigating Or Analyzing Materials By The Use Of Ultrasonic Waves (AREA)
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 invention relates to the technical field of seismic wave forward numerical simulation, in particular to a finite difference template optimization design method of a non-equilateral long-grid wave equation.
Background
The seismic wave forward numerical simulation technology is a process of propagating waves in a complex geological model (including isotropic media, anisotropic media, Biot multiphase anisotropic media, random hole media and the like) under the condition that the complex geological model is known, and receiving the waves by detectors arranged on the earth surface or underground through multiple transmission, reflection and scattering of underground geological structures by using a numerical calculation method. The accurate wave equation numerical solution is utilized to simulate the seismic response of the underground complex geological structure, and a more scientific mathematical physical basis is provided for the research of seismic wave propagation mechanism, special processing method of seismic data, the explanation of complex stratum and other aspects. In recent years, wave equation numerical simulation methods have been widely used in reverse time migration and full waveform inversion.
There are several methods of wave equation forward modeling, the more common being: finite difference methods, pseudo-spectral methods, finite element methods, boundary element methods, spectral element methods, and the like. The finite difference method is widely used because of small calculation amount and high calculation efficiency, and can be suitable for a more complex speed model. The finite difference method can be classified into: explicit finite differences and implicit finite differences; regular grid finite differences, staggered grid finite differences, and rotated staggered grid finite differences. In the finite difference method, the difference coefficient may be obtained by taylor series expansion or an optimization method, and corresponds to a finite difference based on taylor series expansion and a finite difference based on optimization, respectively. In the conventional finite difference method, the difference coefficient is obtained by minimizing the dispersion relation of the spatial domain. In recent years, a time-space domain finite difference method has appeared, which has higher simulation accuracy and better stability by minimizing the dispersion relation between a time domain and a space domain to obtain a difference coefficient.
The current finite difference method of the time-space domain requires equal space sampling intervals in all directions, namely, the model needs to be divided into square or cubic grids. In actual production, in order to meet specific accuracy requirements or to save computation, we often need to subdivide the model into rectangular or rectangular parallelepiped grids (usually, the grid spacing in the depth direction is different from the horizontal direction). The time-space domain finite difference method suitable for square and cube grids cannot meet specific precision requirements or cannot save calculated amount.
Disclosure of Invention
The embodiment of the invention provides a non-equilateral long grid wave equation finite difference template optimization design method, which expands a time-space domain finite difference method only suitable for square and cube grids into rectangular or cuboid grids to meet the specific simulation precision requirement and the requirement of saving calculated amount in actual production, and comprises the following steps:
performing mesh generation on a simulation area of the actual geological model according to the time sampling interval and the space sampling interval;
calculating corresponding operator lengths for different acoustic wave speeds according to the given maximum allowable error and wave number range;
obtaining an optimized finite difference coefficient of the grid based on a time-space domain finite difference method of least square optimization and the corresponding operator length;
and substituting the obtained optimized finite difference coefficient into a differential lattice type wave equation to carry out forward simulation of the wave equation.
In one embodiment, the performing differential mesh generation on the simulation region of the actual geological model according to the time sampling interval and the space sampling interval includes: dividing a simulation area of the actual geological model into a cuboid grid or a rectangular grid;
based on the least square optimization time-space domain finite difference method and the corresponding operator length, the method obtains the optimization finite difference coefficient of the grid, and comprises the following steps: and obtaining the optimized finite difference coefficient of the rectangular grid or the optimized finite difference coefficient of the rectangular grid.
In one embodiment, the optimized finite difference coefficient of the rectangular parallelepiped grid is obtained according to the following formula:
wherein,
where b is the wavenumber, M is the operator length, amIn order to optimize the finite difference coefficient, theta is the included angle between the plane wave propagation direction and the horizontal plane, and theta belongs to [0, pi ]](ii) a Phi is the azimuth angle of plane wave propagation, phi belongs to [0,2 pi ]]；V is the sound wave speed, tau is the time sampling interval, and h is the sampling interval in the x and y directions; c is delta z/h, c is a parameter variable, and delta z is a sampling interval in the z direction; k is a parameter variable, beta is a wave number range, and beta belongs to [0, b ]](ii) a M is a parameter variable, M is an integer, and M belongs to [1, M ∈](ii) a n is a parameter variable, n is an integer, n belongs to [1, M ]]。
In one embodiment, the maximum error corresponding to the optimized finite difference coefficient of the rectangular parallelepiped grid satisfies the following constraint condition:
ξ1max<η；
wherein ξ1maxThe maximum error corresponding to the optimized finite difference coefficient of the cuboid grid is obtained; η is the maximum allowable error of 5.
In one embodiment, the maximum error ξ corresponding to the optimized finite difference coefficient of the cuboid mesh is1maxCalculated according to the following formula:
wherein,
in one embodiment, the performing wave equation forward modeling by substituting the obtained optimized finite difference coefficient into a wave equation in a differential form includes: substituting the obtained optimized finite difference coefficient of the cuboid grid into a three-dimensional sound wave equation of a difference format to carry out forward simulation of the three-dimensional sound wave equation; the three-dimensional sound wave equation of the difference format is as follows:
wherein P is sound pressure. The optimized finite difference coefficient of the rectangular grid is obtained and calculated according to the following formula:
wherein,
where b is the wavenumber, M is the operator length, amIn order to optimize the finite difference coefficient, theta is the included angle between the plane wave propagation direction and the horizontal plane, and theta belongs to [0, pi ]]；V is the acoustic wave speed, tau is the time sampling interval, and h is the sampling interval in the x direction; c is delta z/h, c is a parameter variable, and delta z is a sampling interval in the z direction; k is a parameter variable, beta is a wave number range, and beta belongs to [0, b ]](ii) a M is a parameter variable, M is an integer, and M belongs to [1, M ∈](ii) a n is a parameter variable, n is an integer, n belongs to [1, M ]]。
In one embodiment, the maximum error corresponding to the optimized finite difference coefficient of the rectangular grid satisfies the following constraint condition:
ξ2max<η；
wherein ξ2maxMaximum error corresponding to the optimized finite difference coefficient of the rectangular grid; η is the maximum allowable error.
In one embodiment, the maximum error ξ corresponding to the optimized finite difference coefficient of the rectangular grid2maxCalculated according to the following formula:
wherein, <math>
<mrow>
<msub>
<mi>ξ</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>β</mi>
<mo>,</mo>
<mi>θ</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mi>rβ</mi>
</mfrac>
<mi>arcsin</mi>
<msqrt>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<munderover>
<mi>Σ</mi>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>a</mi>
<mi>m</mi>
</msub>
<mo>[</mo>
<mi>si</mi>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>mβ</mi>
<mn>2</mn>
</mfrac>
<mi>cos</mi>
<mi>θ</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mfrac>
<mi>si</mi>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>mcβ</mi>
<mn>2</mn>
</mfrac>
<mi>sin</mi>
<mi>θ</mi>
<mo>)</mo>
</mrow>
<mo>]</mo>
</msqrt>
<mo>-</mo>
<mn>1</mn>
<mo>.</mo>
</mrow>
</math>
in one embodiment, the substituting the obtained optimized finite difference coefficient into the wave equation in the difference lattice to perform the wave equation forward modeling further includes: substituting the obtained optimized finite difference coefficient of the rectangular grid into a two-dimensional sound wave equation in a difference format to carry out forward simulation of the two-dimensional sound wave equation; the two-dimensional acoustic wave equation of the difference format is as follows:
wherein P is sound pressure.
In the embodiment of the invention, the method for optimally designing the non-equilateral long-mesh wave equation finite difference template expands the time-space domain finite difference method only suitable for square and cubic meshes into rectangular or cuboid meshes, and meets the specific simulation precision requirement and the requirement of saving calculated amount in actual production.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principles of the invention. In the drawings:
FIG. 1 is a flow chart of a finite difference template optimization design method of a non-equilateral long mesh wave equation provided in an embodiment of the present invention;
FIG. 2 is a diagram illustrating the variation of an error curve with propagation direction obtained by a conventional finite difference method according to an embodiment of the present invention;
FIG. 3 is a diagram of the variation of an error curve along with the propagation direction obtained by the finite difference template optimization design method of the non-equilateral long mesh wave equation provided by the embodiment of the present invention;
FIG. 4 is a graph of the variation law of an error curve with operator length obtained by a conventional finite difference method according to an embodiment of the present invention;
FIG. 5 is a diagram of the variation law of an error curve with operator length obtained by a finite difference template optimization design method of a non-equilateral long mesh wave equation provided by the embodiment of the present invention;
fig. 6 is a comparison graph of wave field snapshots at 1.0s of a uniform acoustic wave medium, obtained by a conventional finite difference method and a non-equilateral long mesh wave equation finite difference template optimization design method, respectively, according to an embodiment of the present invention;
FIG. 7 is a comparison of waveforms at the location of the dashed line in FIG. 6;
fig. 8 is a comparison diagram of wave field snapshots at 1.0s of a Marmousi model respectively obtained by a conventional finite difference method and a non-equilateral long mesh wave equation finite difference template optimization design method according to an embodiment of the present invention;
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the following embodiments and accompanying drawings. The exemplary embodiments and descriptions of the present invention are provided to explain the present invention, but not to limit the present invention.
The inventor finds that the current time-space domain finite difference method requires equal space sampling intervals in all directions, namely, a model needs to be divided into square or cubic grids, but the method cannot meet specific precision requirements or can not save calculation amount. If the model is divided into rectangular or cuboid grids, specific precision requirements can be met or calculated amount can be saved. Based on the method, the invention provides a finite difference template optimization design method of a non-equilateral long mesh wave equation.
Fig. 1 is a flowchart of a finite difference template optimization design method for a non-equilateral long mesh wave equation, as shown in fig. 1, the method includes:
step 101: performing mesh generation on a simulation area of the actual geological model according to the time sampling interval and the space sampling interval;
step 102: calculating corresponding operator lengths for different acoustic wave speeds according to the given maximum allowable error and wave number range;
step 103: obtaining an optimized finite difference coefficient of the grid based on a time-space domain finite difference method of least square optimization and the corresponding operator length;
step 104: and substituting the obtained optimized finite difference coefficient into a differential lattice type wave equation to carry out forward simulation of the wave equation.
In specific implementation, the simulation region (or the calculation region) of the actual geological model needs to be divided into rectangular meshes or rectangular grids (usually, the grid spacing in the depth direction is different from the horizontal direction).
The three-dimensional wave equation of sound waves is as follows:
in the formula, P represents a sound pressure, and V represents a sound wave velocity.
Taking the case that the grid distance in the z direction is different from the x and y directions as an example, the specific optimized finite difference format of the rectangular parallelepiped grid is as follows:
wherein τ is a time sampling interval; h is sampling interval in x and y directions; c is delta z/h, c is a parameter variable, and delta z is a sampling interval in the z direction; a ismFor the optimized finite difference coefficient, M represents the operator length; m is a parameter variable, M is an integer, and M belongs to [1, M ∈]。
To simplify notation, we willIs abbreviated asAccording to plane wave theory, we order:
wherein,
wherein i, l, j and n are parameter variables, n is an integer, and n belongs to [1, M ]; k is a parameter variable; theta belongs to [0, pi ] is the included angle between the plane wave propagation direction and the horizontal plane, and phi belongs to [0,2 pi ] is the azimuth angle of the plane wave propagation. Substituting equations (3) and (4) into equation (2), and appropriately simplifying, can obtain:
wherein,then according to the constraint condition;
the following can be obtained:
wherein, beta is kh, beta is wave number range, beta belongs to [0, b ]. The finite difference algorithm is optimized by solving a set of difference coefficients to minimize the error at both ends of equation (7). According to the least squares theory, this problem can be transformed to solve the following system of equations:
wherein,
by solving the system of linear equations (8), we can obtain the optimized difference coefficients of the cuboid grids. And substituting the difference coefficient into the equation (2) to solve the three-dimensional sound wave equation.
In order to ensure the effectiveness of the method of the present invention, the following constraint conditions need to be added in the previous optimization process, that is, the maximum error corresponding to the optimized finite difference coefficient of the rectangular parallelepiped grid satisfies the following constraint conditions:
ξ1max<η (10)
we calculate the maximum error for the optimized difference coefficient by:
wherein,
where η is the maximum allowable error.
ξ1maxOnly with respect to b and M. When M is fixed, xi1maxDependent on b, xi1maxBecomes larger as b increases. Therefore, if the initial b does not satisfy equation (10), we should do so by gradually decreasing b until ξ1max<Eta to obtain the optimum b. In addition, when b is fixed, ξ1maxAssociated with M only, xi1maxBecomes smaller as M increases. Therefore, if the initial M does not satisfy equation (10), we should increase M gradually until ξ1max<η to obtain the optimum M.
For a two-dimensional rectangular grid, taking the case that the grid distance in the z direction is different from that in the x direction as an example, a specific optimized finite difference format is as follows:
wherein P represents sound pressure and V represents sound wave velocity; wherein τ is a time sampling interval; h is a sampling interval in the x direction; c is delta z/h, c is a parameter variable, and delta z is a sampling interval in the z direction; a ismFor limitation after optimizationDifference coefficient, M represents operator length; m is a parameter variable, M is an integer, and M belongs to [1, M ∈]。
We can obtain the optimized difference coefficients for the rectangular grid by solving the following linear equations:
wherein,
by solving the system of linear equations (15), we can obtain the optimized difference coefficients for the rectangular grid. And substituting the difference coefficient into equation (13) to solve the two-dimensional sound wave equation.
Similarly, in order to ensure the effectiveness of the method of the present invention, the following constraint condition needs to be added in the foregoing optimization process, that is, the maximum error corresponding to the optimized finite difference coefficient of the rectangular grid satisfies the following constraint condition:
ξ2max<η (16)
we calculate the maximum error for the optimized difference coefficient by:
wherein,
where η is the maximum allowable error.
The advantages of the present invention are illustrated below by way of example with a uniform acoustic medium. The longitudinal wave velocity is 1500M/s, the time sampling interval is 1ms, the sampling interval in the x direction is 10M, the sampling interval in the z direction is 5M, and the operator length M belongs to [2, 10 ].
The rule of variation of the error curve along with the propagation direction is obtained by using the method of the present invention (i.e. the finite difference template optimization design method of the non-equilateral long mesh wave equation) and the conventional finite difference method, where b is 2.74 and M is 8, as shown in fig. 2 and 3, respectively. Rule eta is 10-2.5And the value is approximately equal to 0.003, and the change rule of the error curve along with the length of an operator is obtained by adopting the method and the traditional finite difference method, which are respectively shown in FIGS. 4 and 5. As can be seen from fig. 2 to 5, compared with the taylor-based finite difference time-space domain method (conventional finite difference method), the method of the present invention has a wider effective frequency band and smaller numerical dispersion. Rule eta is 10-2.5And is approximately equal to 0.003, a wave field snapshot comparison diagram of a uniform acoustic wave medium is obtained by adopting a traditional finite difference method and the method disclosed by the invention, and is shown in figure 6. The left graph in fig. 6 is a wave field snapshot of the uniform acoustic wave medium obtained by the conventional finite difference method, and the right graph is a wave field snapshot of the uniform acoustic wave medium obtained by the method of the present invention. Fig. 7 is a comparison graph of waveforms at the positions of the dotted lines in fig. 6. As can be seen from FIGS. 6 and 7, the simulation accuracy of the method of the present invention is higher when the operator lengths are the same.
To further illustrate the effectiveness of the present invention, we performed forward modeling with a Marmousi model. The time sampling interval is 1ms, the x-direction sampling interval is 12m, and the z-direction sampling interval is 9 m. The operator length M of the conventional finite difference method is 9. The finite difference of the time-space domain adopts an algorithm of changing the length of an operator, operators with different lengths are selected according to different speeds, and M belongs to [2, 9 ]. Fig. 8 is a comparison diagram of wave field snapshots of the Marmousi model at the time of 1.0s, which are respectively obtained by the conventional finite difference method and the method of the present invention, provided in the embodiment of the present invention, the upper graph is a wave field snapshot of the Marmousi model at the time of 1.0s, which is obtained by the conventional finite difference method, and the lower graph is a wave field snapshot of the Marmousi model at the time of 1.0s, which is obtained by the method of the present invention; as can be seen from FIG. 8, the method of the present invention is suitable for forward simulation of complex media, and the simulation effect is significantly better than that of the conventional finite difference method.
In summary, the method of the present invention has the following advantages: 1. the numerical dispersion of the middle and high frequency bands is reduced. 2. The method is more practical based on the time-space domain dispersion relation, and has higher simulation precision. 3. The method is suitable for rectangular and cuboid grids, and can meet specific precision requirements in actual production and save the calculation amount.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and various modifications and changes may be made to the embodiment of the present invention by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (10)
1. A finite difference template optimization design method of a non-equilateral long mesh wave equation is characterized by comprising the following steps:
performing mesh generation on a simulation area of the actual geological model according to the time sampling interval and the space sampling interval;
calculating corresponding operator lengths for different acoustic wave speeds according to the given maximum allowable error and wave number range;
obtaining an optimized finite difference coefficient of the grid based on a time-space domain finite difference method of least square optimization and the corresponding operator length;
and substituting the obtained optimized finite difference coefficient into a differential lattice type wave equation to carry out forward simulation of the wave equation.
2. The method of claim 1, wherein the differentially meshing the simulated region of the actual geological model based on the temporal sampling interval and the spatial sampling interval comprises: dividing a simulation area of the actual geological model into a cuboid grid or a rectangular grid;
based on the least square optimization time-space domain finite difference method and the corresponding operator length, the method obtains the optimization finite difference coefficient of the grid, and comprises the following steps: and obtaining the optimized finite difference coefficient of the rectangular grid or the optimized finite difference coefficient of the rectangular grid.
3. The method of claim 2, wherein the optimized finite difference coefficients of the rectangular parallelepiped grid are obtained according to the following formula:
wherein,
where b is the wavenumber, M is the operator length, amIn order to optimize the finite difference coefficient, theta is the included angle between the plane wave propagation direction and the horizontal plane, and theta belongs to [0, pi ]](ii) a Phi is the azimuth angle of plane wave propagation, phi belongs to [0,2 pi ]]；V is the sound wave speed, tau is the time sampling interval, and h is the sampling interval in the x and y directions; c is delta z/h, c is a parameter variable, and delta z is a sampling interval in the z direction; k is a parameter variable, beta is a wave number range, and beta belongs to [0, b ]](ii) a M is a parameter variable, M is an integer, and M belongs to [1, M ∈](ii) a n is a parameter variableN is an integer, n belongs to [1, M ]]。
4. The method of claim 3, wherein the maximum error corresponding to the optimized finite difference coefficients of the rectangular parallelepiped lattice satisfies the following constraint:
ξ1max<η；
wherein ξ1maxThe maximum error corresponding to the optimized finite difference coefficient of the cuboid grid is obtained; η is the maximum allowable error.
5. The method of claim 4, wherein the optimized finite difference coefficients of the cuboid mesh correspond to a maximum error ξ1maxCalculated according to the following formula:
wherein,
6. the method of claim 5, wherein the substituting the optimized finite difference coefficients into the wave equation in a differential form for wave equation forward modeling comprises: substituting the obtained optimized finite difference coefficient of the cuboid grid into a three-dimensional sound wave equation of a difference format to carry out forward simulation of the three-dimensional sound wave equation; the three-dimensional sound wave equation of the difference format is as follows:
wherein P is sound pressure.
7. The method of claim 2, wherein the optimized finite difference coefficients for the rectangular grid are obtained according to the following formula:
wherein,
where b is the wavenumber, M is the operator length, amIn order to optimize the finite difference coefficient, theta is the included angle between the plane wave propagation direction and the horizontal plane, and theta belongs to [0, pi ]]；V is the acoustic wave speed, tau is the time sampling interval, and h is the sampling interval in the x direction; c is delta z/h, c is a parameter variable, and delta z is a sampling interval in the z direction; k is a parameter variable, beta is a wave number range, and beta belongs to [0, b ]](ii) a m is a parameter variableM is an integer, M is an element of [1, M ]](ii) a n is a parameter variable, n is an integer, n belongs to [1, M ]]。
8. The method of claim 7, wherein the maximum error corresponding to the optimized finite difference coefficients of the rectangular grid satisfies the following constraint:
ξ2max<η；
wherein ξ2maxMaximum error corresponding to the optimized finite difference coefficient of the rectangular grid; η is the maximum allowable error.
9. The method of claim 8, wherein the optimized finite difference coefficients of the rectangular grid correspond to a maximum error ξ2maxCalculated according to the following formula:
wherein, <math>
<mrow>
<msub>
<mi>ξ</mi>
<mn>2</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>β</mi>
<mo>,</mo>
<mi>θ</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mi>rβ</mi>
</mfrac>
<mi>arcsin</mi>
<msqrt>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<munderover>
<mi>Σ</mi>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>a</mi>
<mi>m</mi>
</msub>
<mo>[</mo>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>mβ</mi>
<mn>2</mn>
</mfrac>
<mi>cos</mi>
<mi>θ</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mfrac>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>mcβ</mi>
<mn>2</mn>
</mfrac>
<mi>sin</mi>
<mi>θ</mi>
<mo>)</mo>
</mrow>
<mo>]</mo>
</msqrt>
<mo>-</mo>
<mn>1</mn>
<mo>.</mo>
</mrow>
</math>
10. the method of claim 9, wherein the fitting the optimized finite difference coefficients into a wave equation in a differencing form to perform a wave equation forward simulation, further comprises: substituting the obtained optimized finite difference coefficient of the rectangular grid into a two-dimensional sound wave equation in a difference format to carry out forward simulation of the two-dimensional sound wave equation; the two-dimensional acoustic wave equation of the difference format is as follows:
wherein P is sound pressure.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201510029000.9A CN104597488B (en) | 2015-01-21 | 2015-01-21 | Optimum design method of finite difference template of non-equiangular long-grid wave equation |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201510029000.9A CN104597488B (en) | 2015-01-21 | 2015-01-21 | Optimum design method of finite difference template of non-equiangular long-grid wave equation |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN104597488A true CN104597488A (en) | 2015-05-06 |
| CN104597488B CN104597488B (en) | 2017-05-24 |
Family
ID=53123393
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201510029000.9A Active CN104597488B (en) | 2015-01-21 | 2015-01-21 | Optimum design method of finite difference template of non-equiangular long-grid wave equation |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN104597488B (en) |
Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106646597A (en) * | 2016-12-14 | 2017-05-10 | 中国石油大学(北京) | Forward modeling method and device based on spring network model |
| CN107179549A (en) * | 2017-07-11 | 2017-09-19 | 中海石油(中国)有限公司 | A kind of acoustic wave equation in time domain Explicit finite difference seismic response analogy method |
| CN107976710A (en) * | 2017-11-17 | 2018-05-01 | 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 |
| CN109490955A (en) * | 2018-11-14 | 2019-03-19 | 深圳市勘察研究院有限公司 | A kind of the Acoustic Wave-equation the Forward Modeling and device of rule-based grid |
| CN109490947A (en) * | 2018-10-16 | 2019-03-19 | 中国科学院地质与地球物理研究所 | A kind of high-temperature medium seimic wave propagation analogy method |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102062875A (en) * | 2010-11-30 | 2011-05-18 | 中国石油集团川庆钻探工程有限公司 | Forward modeling method for fluctuating surface elastic wave equation |
| CN102183790A (en) * | 2011-02-12 | 2011-09-14 | 中国石油大学(华东) | Elastic wave forward simulation technology based on space-time dual-variable grid |
| US20120243371A1 (en) * | 2011-03-23 | 2012-09-27 | Chevron U.S.A. Inc. | System and method for seismic data modeling and migration |
| CN103630933A (en) * | 2013-12-09 | 2014-03-12 | 中国石油天然气集团公司 | Nonlinear optimization based time-space domain staggered grid finite difference method and device |
-
2015
- 2015-01-21 CN CN201510029000.9A patent/CN104597488B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102062875A (en) * | 2010-11-30 | 2011-05-18 | 中国石油集团川庆钻探工程有限公司 | Forward modeling method for fluctuating surface elastic wave equation |
| CN102183790A (en) * | 2011-02-12 | 2011-09-14 | 中国石油大学(华东) | Elastic wave forward simulation technology based on space-time dual-variable grid |
| US20120243371A1 (en) * | 2011-03-23 | 2012-09-27 | Chevron U.S.A. Inc. | System and method for seismic data modeling and migration |
| CN103630933A (en) * | 2013-12-09 | 2014-03-12 | 中国石油天然气集团公司 | Nonlinear optimization based time-space domain staggered grid finite difference method and device |
Non-Patent Citations (2)
| Title |
|---|
| 李胜军 等: "声波方程有限差分数值模拟的变网格步长算法", 《工程地球物理学报》 * |
| 金其虎: "弹性波有限差分数值模拟及井间地震逆时偏移成像研究", 《中国优秀硕士学位论文全文数据库·基础科学辑》 * |
Cited By (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106646597A (en) * | 2016-12-14 | 2017-05-10 | 中国石油大学(北京) | Forward modeling method and device based on spring network model |
| CN107179549A (en) * | 2017-07-11 | 2017-09-19 | 中海石油(中国)有限公司 | A kind of acoustic wave equation in time domain Explicit finite difference seismic response analogy method |
| CN107976710A (en) * | 2017-11-17 | 2018-05-01 | 河海大学 | A kind of implicit time-space domain finite difference numerical simulation method of linear optimization based on ACOUSTIC WAVE EQUATION |
| 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 |
| CN109490947A (en) * | 2018-10-16 | 2019-03-19 | 中国科学院地质与地球物理研究所 | A kind of high-temperature medium seimic wave propagation analogy method |
| CN109490947B (en) * | 2018-10-16 | 2020-01-10 | 中国科学院地质与地球物理研究所 | High-temperature medium seismic wave propagation simulation method |
| CN109490955A (en) * | 2018-11-14 | 2019-03-19 | 深圳市勘察研究院有限公司 | A kind of the Acoustic Wave-equation the Forward Modeling and device of rule-based grid |
Also Published As
| Publication number | Publication date |
|---|---|
| CN104597488B (en) | 2017-05-24 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN104597488B (en) | Optimum design method of finite difference template of non-equiangular long-grid wave equation | |
| CN104122585B (en) | Seismic forward simulation method based on elastic wave field resolution of vectors and low-rank decomposition | |
| CN103630933B (en) | Nonlinear optimization based time-space domain staggered grid finite difference method and device | |
| CN107678062B (en) | The integrated forecasting deconvolution of the domain hyperbolic Radon and feedback loop methodology multiple suppression model building method | |
| CN103412328B (en) | Wavenumber domain based on staggering mesh finite-difference algorithm protects amplitude wave field separation method | |
| CN105911584B (en) | Implicit staggered-grid finite difference elastic wave numerical simulation method and device | |
| CN103699798A (en) | Method for realizing numerical simulation of seismic wave field | |
| CN101021568A (en) | Three-dimensional integral prestack depth migration method | |
| CN103149585A (en) | Elastic migration seismic wave field construction method and elastic migration seismic wave field construction device | |
| CN104749628A (en) | Absorbing boundary reflection method based on dispersal viscosity wave equation | |
| CN106443793B (en) | space-time double-variation forward modeling method | |
| CN105676280A (en) | Two-phase medium geological data obtaining method and device based on rotationally staggered grids | |
| CN108279437A (en) | Variable density ACOUSTIC WAVE EQUATION time higher order accuracy staggering mesh finite-difference method | |
| CN103675905A (en) | Optimal coefficient acquisition method and device, and related wave field simulating method and device | |
| CN105182414B (en) | A kind of method that direct wave is removed based on Wave equation forward modeling | |
| CN104850688A (en) | Method for determining elastic ship body load responding model in irregular wave | |
| CN105807317A (en) | Anisotropy attenuation surface wave analogy method based on Chebyshev pseudo-spectral method | |
| CN106054242A (en) | Three dimensional anisotropy attenuation medium wave field simulation method | |
| CN104462814A (en) | Grid grading method for near-surface ground motion simulation | |
| CN105447225A (en) | Combined absorbing boundary condition applied to sound wave finite difference numerical simulation | |
| CN114624766B (en) | Elastic wave least square reverse time migration gradient solving method based on traveling wave separation | |
| CN107807392A (en) | A kind of piecemeal space-time of adaptive anti-frequency dispersion is double to become reverse-time migration method | |
| CN103425881A (en) | Method for certainty numerical simulation of crack medium seismic wave response | |
| CN115270579A (en) | Second-order acoustic wave equation finite difference numerical simulation parameter selection method | |
| CN104750954A (en) | Method and device for simulating earthquake waves in complex anisotropic media |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| C06 | Publication | ||
| PB01 | Publication | ||
| C10 | Entry into substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |