CN109725346A - A kind of time-space high-order VTI rectangular mesh finite difference method and device - Google Patents
A kind of time-space high-order VTI rectangular mesh finite difference method and device Download PDFInfo
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Abstract
This specification embodiment provides a kind of time-space high-order VTI rectangular mesh finite difference method and device.The described method includes: conversion VTI equations for elastic waves is matrix wave equation；The matrix wave equation is expanded to time fourth-order accuracy form, obtains discrete wave equation；Based on the discrete wave equation, space partial derivative is sought；Based on the space partial derivative, optimization finite difference coefficient is sought.By the above method, time discrete precision can effectively improve, it is ensured that wave field extrapolation computational efficiency is improved while precision.
Description
Technical field
This specification embodiment is related to wave field extrapolation technical field, in particular to a kind of time-space high-order VTI rectangle net
Lattice finite difference method and device.
Background technique
It is compared to acoustic wavefield, elastic wave field is capable of providing richer, comprehensive subsurface geological structure, therefore is based on
The migration imaging and waveform inversion of elastic wave wave equation become forward position and the hot spot of seismic prospecting.
In high precision, efficient wave field extrapolation method is one of Elastic Wave Migration imaging and key factor of waveform inversion.
Finite difference method, especially staggered-mesh algorithm, since it realizes simple, calculation amount is small, is easy to the advantages that parallel and wide
It is general to be applied in elastic-wave numerical modeling.When carrying out wave field numerical using finite difference method, elastic wave is influenced
The main reason for equation finite difference discrete precision is the numerical solidification of room and time item.
Mainly have four classes for the High Resolution Finite Difference method of equations for elastic waves at present: the first kind is Spatial higher order, when
Between second order staggered-mesh it is discrete, corresponding difference coefficient is sought by Taylor series expansion, such methods be a little realize letter
It is single, but interfered vulnerable to room and time frequency dispersion；Second class be using linear or nonlinear optimization algorithm to difference coefficient into
Row optimization, spatial accuracy is obviously improved, but time frequency dispersion still has and stability decreases；Third class is to pass through space-time
The dispersion relation in domain obtains the difference coefficient of higher precision, can obtain higher time and spatial simulation precision, realization is compared
Complicated and nonlinear optimization is time-consuming；4th class is by increasing more nets near the paraxonic of traditional cross difference template
Lattice node, to realize time higher order accuracy, time and space can achieve any even-order precision, but realize more multiple
It is miscellaneous, and calculated due to introducing additional grid node, calculation amount increases.
Since space derivation and anisotropic parameters mutually decouple, the sky based on Taylor expansion and optimization difference coefficient
Between high-order discrete scheme can directly be generalized in VTI elastic-wave numerical modeling.In contrast, since time-domain frequency dispersion is
The nonlinear function that difference coefficient and anisotropic parameters intercouple, time-space domain method and passes through and increases paraxonic mesh point
Time higher-order method is difficult to expand in VTI earthquake wave equation numerical simulation.Therefore, at present for the fluctuation in VTI medium
It is discrete that equation still uses time Finite Difference Scheme of Second Order to carry out, and causes in this way when time step is larger, and conventional method can be by
It is even unstable to serious time numerical solidification.
Summary of the invention
The purpose of this specification embodiment is to provide a kind of time-space high-order VTI rectangular mesh finite difference method, with
It realizes under the premise of with degree of precision, efficiently realizes wave field extrapolation using finite difference calculus.
In order to solve the above-mentioned technical problem, it is limited to provide a kind of time-space high-order VTI rectangular mesh for the embodiment of the present application
Difference method and device are achieved in that
A kind of time-space high-order VTI rectangular mesh finite difference method, comprising:
Conversion VTI equations for elastic waves is matrix wave equation；
The matrix wave equation is expanded to time fourth-order accuracy form, obtains discrete wave equation；
Based on the discrete wave equation, space partial derivative is sought；
Based on the space partial derivative, optimization finite difference coefficient is sought.
A kind of time-space high-order VTI rectangular mesh finite difference device, comprising:
Matrix wave equation conversion module is matrix wave equation for converting VTI equations for elastic waves；
Discrete wave equation obtains module and obtains for the matrix wave equation to be expanded to time fourth-order accuracy form
To discrete wave equation；
Space partial derivative seeks module, for being based on the discrete wave equation, seeks space partial derivative；
Finite difference coefficient seeks module, for being based on the space partial derivative, seeks optimization finite difference coefficient.
The technical solution that is there is provided by above this specification embodiment as it can be seen that this specification embodiment by will be in medium
VTI equations for elastic waves continues to convert, when obtaining being conducive to the form of discrete expansion, by the equation expansion to time fourth-order accuracy
Discrete fluctuation equation form seek out space partial derivative therein further according to the discrete wave equation, and for therein
Finite difference coefficient is further sought, to realize for the time-space high-order finite difference method in medium.By upper
Method is stated, more reliably wave field extrapolation can be provided for Elastic Wave Migration imaging and waveform inversion, be conducive to geological prospecting etc.
The progress of operation.
Detailed description of the invention
In order to illustrate more clearly of this specification embodiment or technical solution in the prior art, below will to embodiment or
Attached drawing needed to be used in the description of the prior art is briefly described, it should be apparent that, the accompanying drawings in the following description is only
The some embodiments recorded in this specification, for those of ordinary skill in the art, before not making the creative labor
It puts, is also possible to obtain other drawings based on these drawings.
Fig. 1 is a kind of flow chart of time-space high-order VTI rectangular mesh finite difference method of this specification embodiment；
Fig. 2 is a kind of module map of time-space high-order VTI rectangular mesh finite difference device of this specification embodiment；
Wave field snapshot schematic diagram when Fig. 3 A is 1.4s of the conventional method under the time step of 2ms in this specification；
Wave field snapshot schematic diagram when Fig. 3 B is 2.1s of the conventional method under the time step of 2ms in this specification；
Wave field snapshot schematic diagram when Fig. 3 C is 1.4s of the conventional method under the time step of 1ms in this specification；
Wave field snapshot schematic diagram when Fig. 3 D is 2.1s of the conventional method under the time step of 1ms in this specification；
Wave field snapshot schematic diagram when Fig. 3 E is the 1.4s in this specification one embodiment under the time step of 2ms；
Wave field snapshot schematic diagram when Fig. 3 F is the 2.1s in this specification one embodiment under the time step of 2ms；
Fig. 4 A is the parameter model schematic diagram of velocity of longitudinal wave in this specification one embodiment；
Fig. 4 B is the parameter model schematic diagram of density in this specification one embodiment；
Fig. 4 C is the parameter model schematic diagram of anisotropic parameters in this specification one embodiment；
Fig. 4 D is the parameter model schematic diagram of anisotropic parameters in this specification one embodiment；
Wave field snapshot schematic diagram when Fig. 5 A is 3.6s of the conventional method under the time step of 1.2ms in this specification；
Wave field snapshot schematic diagram when Fig. 5 B is 3.6s of the conventional method under the time step of 0.6ms in this specification；
Wave field snapshot signal when Fig. 5 C is the 3.6s in this specification one embodiment under the time step of 1.2ms
Figure；
Fig. 6 A is that forward modeling of elastic waves earthquake record of the conventional method under the time step of 1.2ms is shown in this specification
It is intended to；
Fig. 6 B is that forward modeling of elastic waves earthquake record of the conventional method under the time step of 0.6ms is shown in this specification
It is intended to；
Fig. 6 C is the forward modeling of elastic waves earthquake record in this specification one embodiment under the time step of 1.2ms
Schematic diagram.
Specific embodiment
Below in conjunction with the attached drawing in this specification embodiment, the technical solution in this specification embodiment is carried out clear
Chu is fully described by, it is clear that described embodiment is only this specification a part of the embodiment, rather than whole implementation
Example.The embodiment of base in this manual, those of ordinary skill in the art are obtained without making creative work
Every other embodiment, all should belong to this specification protection range.
Illustrate a kind of implementation of time-space high-order VTI rectangular mesh finite difference method of the application below in conjunction with attached drawing 1
Example.The executing subject of the method is server.The time-space high-order VTI rectangular mesh finite difference method is specifically real
Apply that steps are as follows:
S100: conversion VTI equations for elastic waves is matrix wave equation.
With the development of address exploration engineering, numerical simulation become in research ball medium the kinematics of seimic wave propagation and
The important means of dynamic characteristic.In all multi-methods of numerical simulation, wave equation numerical method carries out zoning
Discrete grid block describes the wave equation of seimic wave propagation by numerical solution, is a kind of preferable implementation method.It is solving
After obtaining the higher approximate solution of precision of wave equation, the simulation of the propagation to seismic wave can be used to implement.Finite difference calculus
It is to solve for a kind of rapidly and effectively method of wave equation, by carrying out discrete grid block to elastic fluid model, by medium
In wave differential equation be converted into the form of finite difference and solved, realize wave field extrapolation.
Isotropic medium (isotropic medium) refers to substance performance (physics, chemistry etc. in a different direction
Performance) identical medium.In isotropic medium, the spread speed of light and the polarization direction of light are unrelated, i.e., only have to light
A kind of refractive index.Empty gas and water, optical glass, simple glass etc. are isotropic mediums.VTI medium, i.e. transverse isotropy
(Vertical Transversely Isotropy) medium is one of isotropic medium, only horizontally has phase
Same physical property.In VTI medium, elastic wave can be by the state parameter of elastic wave velocity therein, stress and medium
Etc. equation composed by information be indicated.According to the equations for elastic waves, by a series of conversions, may be implemented based on rectangle
The time-space high-order finite difference method of staggered-mesh.
It is continued below specifically to analyze.For the elastic wave in VTI medium, there are the equations of following form:
Wherein, ρ is the density of medium, (vx,vz) it is velocity component, (τxx,τzz,τxz) it is the components of stress,WithFor VTI elasticity
Figure parameters, wherein vpAnd vsRespectively longitudinal wave and shear wave velocity, ε and δ are Thomsen anisotropic parameters.Above-mentioned equation master
The property of speed therein and stress is described, elastic wave is specifically mapped on corresponding state parameter, so that for
Solution procedure later is more intuitive, and subsequent process is better achieved.
For the ease of the deduction after being carried out to velocity component therein and the components of stress, and also to solution later
It is more convenient, the first matrix p is set to velocity component (vx,vz) and the components of stress (τxx,τzz,τxz) integrated, even p=
(vx,vz,τxx,τzz,τxz)T.So, for the equations for elastic waves in above-mentioned VTI medium, it can use the first matrix to it
It is indicated, the solution procedure after facilitating.
In the case where having set the first matrix, can more easily to the equations for elastic waves in VTI medium into
Row indicates.Based on the first set matrix p=(vx,vz,τxx,τzz,τxz)T, by above-mentioned equations for elastic wavesIt is converted into matrix wave equationWherein, w is set as the second matrix,ρ is the density of medium, (vx,vz) it is velocity component, (τxx,τzz,
τxz) it is the components of stress,WithFor VTI coefficient of elasticity parameter, vpAnd vsRespectively longitudinal wave and shear wave velocity,
ε and δ is Thomsen anisotropic parameters.
S200: the matrix wave equation is expanded to time fourth-order accuracy form, obtains discrete wave equation.
It, will be about the first matrix according to rectangle staggered-mesh in order to realize the discrete expression for the elastic wave in medium
Equations for elastic waves expand into discrete form, thus later can to elastic wave field realize extrapolate.Specifically show at one
In example, at timing node t+ Δ t matrix wave equation using Taylor series being carried out expansion, to reach time quadravalence discrete
Precision obtains discrete wave equationIn formula,Wherein,
ρ is density,
WithFor coefficient of elasticity parameter, vpFor
Velocity of longitudinal wave, vsFor shear wave velocity, ε and δ are Thomsen anisotropic parameters.
Wave field extrapolation can effectively be realized according to the state parameter of known region based on the discrete wave equation, to working as
Preceding target measurement region carries out digital simulation.
S300: it is based on the discrete wave equation, seeks space partial derivative.
In order to further improve discrete wave equation, it is still desirable to calculate space partial derivative therein.
For the single order space partial derivative in the second matrixWithThe discrete scheme of single order space partial derivative can indicate
For following form:
Wherein, hxWith hzRespectively represent the space interval along x and z directions, N1For single order Space Operators length, cnHave for single order
Limit difference coefficient.
For three rank space partial derivatives in third matrixWithThe discrete scheme of three rank space partial derivatives can be with
It is expressed as form:
In formula, hxWith hzRespectively represent the space interval along x and z directions, N2For Space Operators length, cn' it is three rank finite differences
Divide coefficient.
For the three rank space partial derivative of mixing in third matrixWithMix three rank space partial derivatives from
Scattered format can be expressed as form:
By the above-mentioned three rank space partial derivatives soughtWithResult of seeking bring back to
In three matrixes, the third matrix can be sought out.
It is consistent with points used in the partial derivative of single order space are calculated to calculate points used when three rank space partial derivatives.
Using more points, even higher simulation precision can be obtained when Space Operators equal length.But it is empty with single order is calculated
Between partial derivative high-order it is discrete similar, when calculating three rank space partial derivatives use biggish Space Operators length Shi Huixian
It writes and increases additional calculation amount.Therefore, in actual calculating, for EQUILIBRIUM CALCULATION FOR PROCESS efficiency and precision, using formula N2=int
(N1/ 2) three rank space partial derivatives are calculated, the efficiency of calculating is promoted under the premise of ensuring precision.
S400: being based on the space partial derivative, seeks optimization finite difference coefficient.
The optimization finite difference coefficient, including first-order difference coefficient and third order difference coefficient.
For the first-order difference coefficient c in the partial derivative of single order spacen(n=1,2 ..., N1), it can be by solving linear side
Journey group
It is sought.After first-order difference coefficient is calculated, bring into
In the discrete scheme of single order space partial derivative, then by single order space partial derivativeWithIt brings second matrix into, can be obtained
The second matrix that can be specifically sought, convenient for calculating later.
For the third order difference coefficient c ' in three rank space partial derivativesn(n=1,2 ..., N2), it can be linear by solving
Equation groupIt is sought.Above-mentioned equation is solved, will be obtained
Third order difference coefficient bring three rank partial derivatives intoWithThird matrix is sought after can be used to.
The above-mentioned difference coefficient sought based on Taylor expansion can obtain higher essence in small wavenumber range
Degree, but with the increase of wave number, frequency dispersion error can also be gradually increased.It, can be using most in order to further increase simulation precision
Small least square method optimizes finite difference coefficient.In conjunction with plane wave solution equation and the space partial derivative, third side is obtained
Cheng Hou, intermediate equation both ends error described in minimization, and least squares theory is combined, after obtaining optimization method, it is carried out
It solves, finally obtains the finite difference coefficient of optimization.
The three rank staggered-mesh difference coefficients that a specific embodiment is used to seek optimization are described below.By plane wave solutionIt is brought into equationIn, it is available after abbreviationWherein, hxWith hzRespectively represent the space interval along x and z directions, N2For Space Operators length, κn
(β)=2sin [(n-1/2) β], f (β)=β3, β=kxhx, β ∈ [0, π], kx、kzFor x and z directions space wave number.Minimization side
JourneyThe error at both ends, it is as follows to establish objective function:
Based on Least Square Theory, obtaining optimization method is
The optimization method is solved using least square method again, the excellent of three rank space partial derivatives can be obtained
The finite difference coefficient of change.
Described method based on the above embodiment can equally obtain dividing first difference seeking for coefficient, then
It is secondary no longer to repeat one by one.Using the finite difference coefficient of the optimization, preferably experimental result can be sought, it is more smart
Really obtain elastic wave field analog result.
Using the above method into specific embodiment, and the comparison between the effect realized with conventional method is combined, card
The validity and accuracy of bright the application.
In one embodiment, as shown in Fig. 3 A to 3F, for uniform dielectric model, the dimension of setting model is 800 ×
800, the grid spacing along the direction x is 15m, and the grid spacing along the direction z is 10m.The speed of P wave and S wave is respectively
3100m/s and 1750m/, Media density 2300kg/m3, anisotropic parameters value is respectively ε=0.15, δ=- 0.15.Dominant frequency
The generation vibration of model center is placed in for the Ricker wavelet of 27Hz.For traditional staggering mesh finite-difference method, using space
Operator length is N1=8.For improved time quadravalence staggering mesh finite-difference method, use Space Operators length for N1=8
And N2=4.Fig. 3 A is the wave field snapshot schematic diagram that conventional method is calculated using the time step of 2ms in 1.4s, Fig. 3 C
For the wave field snapshot schematic diagram that conventional method is calculated using the time step of 1ms in 1.4s, Fig. 3 E is according to the application
The wave field snapshot schematic diagram that the method is calculated using the time step of 2ms in 1.4s.Shown by comparing above three
Intention can be seen that conventional method accuracy when use is compared with large time step and be substantially reduced, and when the same time step of use,
The schematic diagram that the application is obtained is more accurate, obtained experiment effect when having reached conventional method using smaller time step
Fruit.In the case where realizing same accuracy, herein described method can be calculated using bigger time step, in this way
One can undoubtedly reduce calculating step, to improve computational efficiency.Fig. 3 B be conventional method use 2ms time step when
The wave field snapshot schematic diagram being calculated when 2.1s, Fig. 3 D are calculated when being time step of the conventional method using 1ms in 2.1s
Obtained wave field snapshot schematic diagram, Fig. 3 F are wave when using the time step of 2ms according to herein described method in 2.1s
Field snapshot schematic diagram.By above three schematic diagram, the accuracy and validity of the application are further demonstrated.
In order to further verify effect acquired by the application, numerical simulation survey is carried out using two-dimentional Hess VTI model
Examination.Fig. 4 A is the parameter model schematic diagram of velocity of longitudinal wave in the Hess VTI model, and Fig. 4 B is in the Hess VTI model
The parameter model schematic diagram of density, Fig. 4 C are that the parameter model of Thomsen anisotropic parameters ε in the Hess VTI model shows
It is intended to, Fig. 4 D is the parameter model schematic diagram of Thomsen anisotropic parameters δ in the Hess VTI model.The model
It by the model resampling is 1808 × 750 grid cells using the grid of 12m × 12m having a size of 22000m × 9000m.Edge
The direction x mesh spacing be 10m, along the direction z mesh spacing be 15m.The speed ratio of longitudinal and shear wave is 1.7:1, and focus is
The Ricker wavelet of a cycle, dominant frequency 12Hz, hypocentral location are (11.28km, 0.12km), when record a length of 6s.For passing
The staggering mesh finite-difference method of system, uses Space Operators length for N1=12.For improved time quadravalence staggered-mesh
Finite difference method is respectively N using Space Operators length1=12 and N2=6.Fig. 5 A is the time that conventional method uses 1.2ms
The forward modeling of elastic waves wave field snapshot schematic diagram being calculated when step-length in 3.6s, Fig. 5 B are that conventional method uses 0.6ms
Time step when the forward modeling of elastic waves wave field snapshot schematic diagram that is calculated in 3.6s, Fig. 5 C is the embodiment of the present application
Using 1.2ms time step when the forward modeling of elastic waves wave field snapshot schematic diagram that is calculated in 3.6s.By above-mentioned
Comparison between three schematic diagrames can be seen that conventional method can improve the precision of itself by reducing time step, but
It is that the accuracy for the wave field that itself is calculated can not be guaranteed when time step is larger, is easy by time frequency dispersion
Interference.But in the effect shown according to the embodiment that herein described method obtains, it can be seen that using identical
When time step, wave field snapshot acquired in the embodiment of the present application can have the not long gained of smaller time in traditionally method
The precision arrived, that is, the method that the application is taken can use smaller time step in identical intensive qualifications
It is long, to reduce calculation amount, it is able to promote computational efficiency.Further, Fig. 6 A is the time step that conventional method uses 1.2ms
The forward modeling of elastic waves earthquake record schematic diagram that chronistor obtains, Fig. 6 B are the time step that conventional method uses 0.6ms
When the forward modeling of elastic waves earthquake record schematic diagram that is calculated, Fig. 6 C be herein described method using 1.2ms when
The forward modeling of elastic waves earthquake record schematic diagram that spacer step chronistor obtains.By the comparison between above three schematic diagram,
Further illustrate the accuracy and reliability of herein described method.
A kind of embodiment of time-space high-order VTI rectangular mesh finite difference device of the application introduced below.Such as Fig. 2
Shown, which includes:
Matrix wave equation conversion module 210 is matrix wave equation for converting VTI equations for elastic waves；
Discrete wave equation obtains module 220, for the matrix wave equation to be expanded to time fourth-order accuracy form,
Obtain discrete wave equation；
Space partial derivative seeks module 230, for being based on the discrete wave equation, seeks space partial derivative；
Optimization finite difference coefficient seeks module 240, for being based on the space partial derivative, seeks optimization finite difference system
Number.
The matrix wave equation conversion module 210, comprising:
Subelement 211 is arranged in parameter matrix, for based on the velocity component and the components of stress in equations for elastic waves, setting ginseng
Matrix number p=(vx,vz,τxx,τzz,τxz)T, wherein vx,vzFor the velocity component of elastic wave, τxx,τzz,τxzFor answering for elastic wave
Force component；
Matrix wave equation obtains subelement 212, is used for equations for elastic wavesIt is converted into
Matrix wave equationWherein, w is the second matrix,ρ is
Density,WithFor bullet
Property coefficient parameter, vpFor velocity of longitudinal wave, vsFor shear wave velocity, ε and δ are Thomsen anisotropic parameters.
The discrete wave equation obtains module 220, comprising:
Discrete fluctuation equation expansion subelement 221, for utilizing Taylor series expansion, by matrix wave equation
Expand into the discrete wave equation of time fourth-order accuracyIn formula,Wherein,
ρ is density,
WithFor coefficient of elasticity parameter, vpFor
Velocity of longitudinal wave, vsFor shear wave velocity, ε and δ are Thomsen anisotropic parameters.
The space partial derivative seeks module 230, comprising:
Single order space partial derivative seeks subelement 231, for utilizing formula
Seek single order space partial derivativeWithIn formula, hxWith hzRespectively along the space interval of x and z directions, N1For single order Space Operators length, cnHave for single order
Limit difference coefficient；
Three rank space partial derivatives seek subelement 232, for utilizing formula
Seek three rank space partial derivativesWithIn formula, hxWith hzRespectively represent the space interval along x and z directions, N2For three rank Space Operators length, cn' be
Three rank finite difference coefficients；
It mixes three rank space partial derivatives and seeks subelement 233, for utilizing formula
Seek three rank space partial derivatives of mixing, in formula, hxWith hzRespectively represent the space interval along x and z directions, N2For three ranks
Space Operators length.
The optimization finite difference coefficient, including first difference divide coefficient and three rank finite difference coefficients；The optimization
Finite difference coefficient seeks module 240, comprising:
First difference divides coefficient to seek subelement 241, for being based on single order space partial derivative, is sought using Taylor expansion
First-order systemIt obtains first difference and divides coefficient cn(n=1,
2,...,N1), in formula, N1For Space Operators length；
Three rank finite difference coefficients seek subelement 242, for being based on three rank space partial derivatives, are sought using Taylor expansion
Third-order equation groupObtain three rank finite difference coefficient c 'n(n=1,
2,...,N2), in formula, N2For Space Operators length.
Meanwhile the optimization finite difference coefficient seeks module 240, can also include:
Intermediate equation seeks subelement 243, for obtaining intermediate equation in conjunction with plane wave solution and the space partial derivative；
Optimization method seeks subelement 244, for equation both ends error among minimization, and combines least square method, obtains
To optimization method；
Finite difference coefficient computation subunit 245 obtains optimization finite difference system for solving to optimization method
Number.
The finite difference coefficient as described in limited difference coefficient computation subunit 245 comprises at least one of the following:
First difference divides coefficient；
Three rank finite difference coefficients.
In the 1990s, the improvement of a technology can be distinguished clearly be on hardware improvement (for example,
Improvement to circuit structures such as diode, transistor, switches) or software on improvement (improvement for method flow).So
And with the development of technology, the improvement of current many method flows can be considered as directly improving for hardware circuit.
Designer nearly all obtains corresponding hardware circuit by the way that improved method flow to be programmed into hardware circuit.Cause
This, it cannot be said that the improvement of a method flow cannot be realized with hardware entities module.For example, programmable logic device
(Programmable Logic Device, PLD) (such as field programmable gate array (Field Programmable Gate
Array, FPGA)) it is exactly such a integrated circuit, logic function determines device programming by user.By designer
Voluntarily programming comes a digital display circuit " integrated " on a piece of PLD, designs and makes without asking chip maker
Dedicated IC chip 2.Moreover, nowadays, substitution manually makes IC chip, and this programming is also used instead mostly
" logic compiler (logic compiler) " software realizes that software compiler used is similar when it writes with program development
Seemingly, and the source code before compiling also handy specific programming language is write, this is referred to as hardware description language
(Hardware Description Language, HDL), and HDL is also not only a kind of, but there are many kind, such as ABEL
(Advanced Boolean Expression Language)、AHDL(Altera Hardware Description
Language)、Confluence、CUPL(Cornell University Programming Language)、HDCal、JHDL
(Java Hardware Description Language)、Lava、Lola、MyHDL、PALASM、RHDL(Ruby
Hardware Description Language) etc., VHDL (Very-High-Speed is most generally used at present
Integrated Circuit Hardware Description Language) and Verilog2.Those skilled in the art
It will be apparent to the skilled artisan that only needing method flow slightly programming in logic and being programmed into integrated circuit with above-mentioned several hardware description languages
In, so that it may it is readily available the hardware circuit for realizing the logical method process.
System, device, module or the unit that above-described embodiment illustrates can specifically realize by computer chip or entity,
Or it is realized by the product with certain function.It is a kind of typically to realize that equipment is computer.Specifically, computer for example may be used
Think personal computer, laptop computer, cellular phone, camera phone, smart phone, personal digital assistant, media play
It is any in device, navigation equipment, electronic mail equipment, game console, tablet computer, wearable device or these equipment
The combination of equipment.
As seen through the above description of the embodiments, those skilled in the art can be understood that this specification
It can realize by means of software and necessary general hardware platform.Based on this understanding, the technical solution of this specification
Substantially the part that contributes to existing technology can be embodied in the form of software products in other words, the computer software
Product can store in storage medium, such as ROM/RAM, magnetic disk, CD, including some instructions are used so that a computer
Equipment (can be personal computer, server or the network equipment etc.) executes each embodiment of this specification or embodiment
Certain parts described in method.
All the embodiments in this specification are described in a progressive manner, same and similar portion between each embodiment
Dividing may refer to each other, and each embodiment focuses on the differences from other embodiments.Especially for system reality
For applying example, since it is substantially similar to the method embodiment, so being described relatively simple, related place is referring to embodiment of the method
Part explanation.
This specification can be used in numerous general or special purpose computing system environments or configuration.Such as: personal computer,
Server computer, handheld device or portable device, laptop device, multicomputer system, microprocessor-based system,
Set top box, programmable consumer-elcetronics devices, network PC, minicomputer, mainframe computer including any of the above system are set
Standby distributed computing environment etc..
This specification can describe in the general context of computer-executable instructions executed by a computer, such as journey
Sequence module.Generally, program module include routines performing specific tasks or implementing specific abstract data types, programs, objects,
Component, data structure etc..This specification can also be practiced in a distributed computing environment, in these distributed computing environment
In, by executing task by the connected remote processing devices of communication network.In a distributed computing environment, program module
It can be located in the local and remote computer storage media including storage equipment.
Although depicting this specification by embodiment, it will be appreciated by the skilled addressee that there are many become for this specification
Shape and the spirit changed without departing from this specification, it is desirable to which the attached claims include these deformations and change without departing from this
The spirit of specification.
Claims (14)
1. a kind of time-space high-order VTI rectangular mesh finite difference method characterized by comprising
Conversion VTI equations for elastic waves is matrix wave equation；
The matrix wave equation is expanded to time fourth-order accuracy form, obtains discrete wave equation；
Based on the discrete wave equation, space partial derivative is sought；
Based on the space partial derivative, optimization finite difference coefficient is sought.
2. the method as described in claim 1, which is characterized in that the conversion VTI equations for elastic waves is matrix wave equation, packet
It includes:
Based on the velocity component and the components of stress in VTI equations for elastic waves, parameter matrix p=(v is setx,vz,τxx,τzz,τxz)T,
Wherein, vx,vzFor the velocity component of elastic wave, τxx,τzz,τxzFor the components of stress of elastic wave；
By equations for elastic wavesIt is converted into matrix wave equationWherein, w is the second square
Battle array,ρ is density, WithFor coefficient of elasticity parameter, vpFor velocity of longitudinal wave, vsFor shear wave
Speed, ε and δ are Thomsen anisotropic parameters.
3. the method as described in claim 1, which is characterized in that described that the matrix wave equation is expanded to time quadravalence essence
Degree form obtains discrete wave equation, comprising:
Using Taylor series expansion, by matrix wave equationExpand into the discrete wave equation of time fourth-order accuracyIn formula,
Wherein,
ρ is density,
WithFor coefficient of elasticity parameter, vpFor longitudinal wave
Speed, vsFor shear wave velocity, ε and δ are Thomsen anisotropic parameters.
4. the method as described in claim 1, which is characterized in that it is described to be based on the discrete wave equation, seek space local derviation
Number, comprising:
Utilize formula Seek single order space partial derivativeWith
In formula, hxWith hzRespectively along the space interval of x and z directions, N1For single order Space Operators length, cnFor first difference point
Coefficient；
Alternatively, utilizing formula Seek three rank space partial derivativesWithIn formula, hxWith hzRespectively represent the space interval along x and z directions, N2For three rank Space Operators length, cn' have for three ranks
Limit difference coefficient；
Utilize formula It is empty to seek three ranks of mixing
Between partial derivative, in formula, hxWith hzRespectively represent the space interval along x and z directions, N2For three rank Space Operators length.
5. the method as described in claim 1, which is characterized in that the optimization finite difference coefficient, including first difference point
Coefficient and three rank finite difference coefficients；It is described to be based on the space partial derivative, seek optimization finite difference coefficient, comprising:
Based on single order space partial derivative, first-order system is sought using Taylor expansionIt obtains first difference and divides coefficient cn(n=1,2 ..., N1), in formula,
N1For Space Operators length；
Alternatively, being based on three rank space partial derivatives, third-order equation group is sought using Taylor expansionObtain three rank finite difference coefficient c 'n(n=1,2 ..., N2), formula
In, N2For Space Operators length.
6. the method as described in claim 1, which is characterized in that it is described to be based on the space partial derivative, seek optimization finite difference
Divide coefficient, comprising:
In conjunction with plane wave solution and the space partial derivative, intermediate equation is obtained；
Equation both ends error among minimization, and least square method is combined, obtain optimization method；
Optimization method is solved, optimization finite difference coefficient is obtained.
7. method as claimed in claim 6, which is characterized in that the optimization finite difference coefficient comprises at least one of the following:
First difference divides coefficient；
Three rank finite difference coefficients.
8. a kind of time-space high-order VTI rectangular mesh finite difference device, which is characterized in that including；
Matrix wave equation conversion module is matrix wave equation for converting VTI equations for elastic waves；
Discrete wave equation obtains module, for the matrix wave equation to be expanded to time fourth-order accuracy form, obtain from
Dissipate wave equation；
Space partial derivative seeks module, for being based on the discrete wave equation, seeks space partial derivative；
Optimization finite difference coefficient seeks module, for being based on the space partial derivative, seeks optimization finite difference coefficient.
9. device as claimed in claim 8, which is characterized in that the matrix wave equation conversion module, comprising:
Subelement is arranged in parameter matrix, for parameter matrix p to be arranged based on the velocity component and the components of stress in equations for elastic waves
=(vx,vz,τxx,τzz,τxz)T, wherein vx,vzFor the velocity component of elastic wave, τxx,τzz,τxzFor the components of stress of elastic wave；
Matrix wave equation obtains subelement, is used for equations for elastic wavesIt is converted into matrix fluctuation
EquationWherein, w is the second matrix,ρ is density,WithFor coefficient of elasticity
Parameter, vpFor velocity of longitudinal wave, vsFor shear wave velocity, ε and δ are Thomsen anisotropic parameters.
10. device as claimed in claim 8, which is characterized in that the discrete wave equation obtains module, comprising:
Discrete fluctuation equation expansion subelement, for utilizing Taylor series expansion, by matrix wave equationWhen expanding into
Between fourth-order accuracy discrete wave equationIn formula,Wherein,
ρ is density,
WithFor coefficient of elasticity parameter, vpFor longitudinal wave
Speed, vsFor shear wave velocity, ε and δ are Thomsen anisotropic parameters.
11. device as claimed in claim 8, which is characterized in that the space partial derivative seeks module, comprising:
Single order space partial derivative seeks subelement, for utilizing formula
Seek single order space partial derivativeWithIn formula, hxWith hzRespectively along the space interval of x and z directions, N1For single order Space Operators length, cnFor first difference
Divide coefficient；
Three rank space partial derivatives seek subelement, for utilizing formula
Seek three rank space partial derivatives
WithIn formula, hxWith hzRespectively represent the space interval along x and z directions, N2For three rank Space Operators length, c 'nFor three ranks
Finite difference coefficient；
It mixes three rank space partial derivatives and seeks subelement, for utilizing formula
Seek mixing three
Rank space partial derivative, in formula, hxWith hzRespectively represent the space interval along x and z directions, N2For three rank Space Operators length.
12. device as claimed in claim 8, which is characterized in that the optimization finite difference coefficient, including first difference point
Coefficient and three rank finite difference coefficients；The optimization finite difference coefficient seeks module, comprising:
First difference divides coefficient to seek subelement, for being based on single order space partial derivative, seeks single order side using Taylor expansion
Journey groupIt obtains first difference and divides coefficient cn(n=1,2 ..., N1),
In formula, N1For Space Operators length；
Three rank finite difference coefficients seek subelement, for being based on three rank space partial derivatives, seek three rank sides using Taylor expansion
Journey groupObtain three rank finite difference coefficient c 'n(n=1,2 ...,
N2), in formula, N2For Space Operators length.
13. device as claimed in claim 8, which is characterized in that the optimization finite difference coefficient seeks module, comprising:
Intermediate equation seeks subelement, for obtaining intermediate equation in conjunction with plane wave solution and the space partial derivative；
Optimization method seeks subelement, for equation both ends error among minimization, and combines least square method, obtains optimization side
Journey；
Optimize finite difference coefficient computation subunit, for solving to optimization method, obtains optimization finite difference coefficient.
14. device as claimed in claim 13, which is characterized in that the optimization finite difference coefficient includes following at least one
Kind:
First difference divides coefficient；
Three rank finite difference coefficients.
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