CN106814390A - Staggered-mesh the Forward Modeling based on time-space domain optimization - Google Patents
Staggered-mesh the Forward Modeling based on time-space domain optimization Download PDFInfo
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Abstract
The invention discloses the staggered-mesh the Forward Modeling optimized based on time-space domain, the method obtains its plane wave general solution to single order pressure-speed partial differential equations first, obtain the analytical expression of the pressure field and polarization velocity field in single order ACOUSTIC WAVE EQUATION, then traditional spatial intersecting grid difference form is combined with precise time recurrence algorithm, obtain accurate staggered time recurrence algorithm, finally solved using least-squares iteration, obtain high accuracy staggered-mesh difference operator for forward simulation.The present invention improves numerical solidification to a certain extent, improves phase velocity precision, and significant effect can be especially reached in the case of exponent number high.
Description
Technical field
It is a kind of staggered-mesh the Forward Modeling based on time-space domain optimization the invention belongs to geophysics treatment technology.
Background technology
With going deep into for Seismic exploration and development, to the requirement such as Forward Problem of Vsp precision, subsurface imaging result resolution ratio
More and more higher, reverse-time migration imaging, least square reverse-time migration and the full waveform inversion imaging based on wave field extrapolation are
Focus as research, the efficiency and precision of wave field extrapolation directly affects the efficiency and precision of follow-up imaging and inverting.It is limited
Calculus of finite differences takes into account computational efficiency and simulation precision, in having been widely used for the wave field extrapolation of seismic exploration at present.Finite difference
Point-score approaches continuous partial derivative operator using discrete difference operator, generally there is numerical solidification in the calculation, influences earthquake
Wave field simulation precision.
The Taylor expansion that conventional finite difference coefficient is based at zero wave number is asked for, and relatively accurately can simulate ground in lower wave number section
Seismic wave propagation, but serious numerical solidification occurs in wave number high section.How to improve difference operator is to suppress numerical solidification
The key areas of finite-difference forward modeling research.Currently for numerical solidification, many researchers have carried out largely both at home and abroad
Research work, conventional strategy mainly includes：(1) the room and time sampling interval is reduced；(2) time or sky are increased
Between difference order；(3) flux-corrected transport method (FCT).Such as the patent of Publication No. patented technology CN103823239A
A kind of frequency domain optimization mixing staggered-mesh finite difference analogy method is disclosed, steps of the method are：1. be given
Time-domain two dimension ACOUSTIC WAVE EQUATION；2. Artificial Boundaries reflection is eliminated, the time-domain two dimension sound with completely permutation boundary condition is obtained
Wave equation；3. Fourier transform is carried out to equation the right and left time variable and obtains frequency domain ACOUSTIC WAVE EQUATION；4. to matching layer side
Boundary conditional frequency domain ACOUSTIC WAVE EQUATION routinely staggered-mesh to carry out finite difference discrete, obtain finite difference discrete scheme；5. it is right
Matching layer boundary condition frequency domain ACOUSTIC WAVE EQUATION by rotationally staggered grid carry out finite difference it is discrete the discrete lattice of finite difference
Formula；6. by conventional staggered-mesh and rotationally staggered grid optimization mixing, with the weighted average in two nested grid systems, quality
Point and 8 points of weighted average around it centered on acceleration term；7. under the minimum criterion of phase velocity error, ask for optimizing
Coefficient.The invention weight coefficient reduces the discrete frequency dispersion error for causing of finite difference, improves frequency domain forward simulation
Precision.Though these three methods improve numerical solidification to a certain extent, the increase of amount of calculation has been resulted in.At present,
Computationally intensive is still restriction three-dimensional high-precision exploration, the least-squares migration based on inverting thought, full waveform inversion, respectively
The key factor of the popularization and application such as anisotropy correlative study.Temporal and spatial sampling interval is either reduced, difference order is improved, also
It is FCT corrections, it is all more obvious in the practical middle advantage of theoretical method and deficiency.
In addition, the research for just drilling optimization about finite difference at present is built upon greatly on the basis of second order ACOUSTIC WAVE EQUATION, to one
Rank variable density ACOUSTIC WAVE EQUATION research is less.However, both there is the change that velocity variations there is also density in true underground medium,
A host of facts show that single order pressure-rate equation is more favorable for processing variable density medium.
The content of the invention
Numerical solidification can be effectively suppressed present invention aim at one kind is provided, and improves earthquake group velocity to a certain extent
The staggered-mesh the Forward Modeling of computational accuracy, the method with the numerical simulation of single order ACOUSTIC WAVE EQUATION group as research object,
The analytical expression of the pressure field and polarization velocity field in single order ACOUSTIC WAVE EQUATION is derived, is given accordingly a kind of high-precision
Recurrence algorithm, asks for matching factor, it is contemplated that the pathosis that coefficient is solved using least square method in Effective wave number section, adopts
With the conjugate gradient iterative algorithm of pretreatment, when variable density medium is processed, numerical solidification can be preferably suppressed, while can
To improve group velocity precision.Specific method of the invention is as follows.
A kind of staggered-mesh the Forward Modeling based on time-space domain optimization, comprises the following steps：
Step 1：Single order pressure-speed partial differential equations form is given, its plane wave general solution is obtained, single order sound wave is obtained
Pressure field and the analytical expression for polarizing velocity field in equation, wherein, the analytical expression of speed is polarized with pressure field
Analytic solutions are represented；
Step 2：Spatial intersecting grid difference form is combined with time recurrence algorithm, plane wave general solution is brought into, is handed over
Stagger the time a recurrence algorithm；
Step 3：Staggered-mesh difference operator is solved using least-squares iteration in wave-number domain；
Step 4：The difference operator that will optimize is used for Acoustic Forward Modeling.
Further, in step 1, the partial differential equation to single order pressure-speed partial differential equations are sat by Descartes
Utilization space integral transformation obtains ODE in mark system；Time integral is done to this ODE equation and obtains plane wave
General solution；Two-dimensional space Fourier transformation and time integral are done respectively to equation both sides and constant term is omitted, obtains polarizing speed
The general solution of component；Here, the analytical expression of polarization speed is represented by the analytic solutions of pressure field.
Further, in step 2, with space difference be combined together time difference by staggered time recurrence algorithm, brings into
Plane wave general solution, respectively obtains the response of time difference and space difference in wave-number domain, will analytic solutions bring spatial intersecting into
Grid difference form, obtains the accurate response of time difference and space difference in wave-number domain, and target is with the letter on the right of equation
Number (space difference filter response) is accurately solved in (the time difference filter response) that wave-number domain approaches the equation left side, most
Time difference is realized eventually with space difference while reaching high accuracy.Due to polarizing the analytical expression of speed by pressure field
Analytic solutions are represented, it is contemplated that time difference and content of both the difference of space, so the staggered time recurrence algorithm will
It refers to bring analytic solutions into spatial intersecting grid difference form that time difference is combined with space difference, and the recurrence algorithm for obtaining is i.e.
It is staggered time recurrence algorithm.
Further, in step 3, target is to approach the equation left side in wave-number domain with the function on the right of equation accurately to solve,
Matching factor is changed by objective function so that matching factor is approached accurately as far as possible in certain wave-number range
Time Recursion Operator；Discrete sampling at equal intervals is carried out to object function equation, coefficient is asked for that to be converted to least square excellent
Change problem is simultaneously iterated solution, obtains high accuracy staggered-mesh difference operator.
Further, in step 3, using time interleaving difference operator wave number domain response as object function, and space is used
Difference operator wave number domain response carries out least square approximation as basic function, and problem can be summarized as least squares problem,
Conjugate gradient iterative solution method using pretreatment increases the stability of solution.
Further, in step 3, being solved using least-squares iteration is included：By staggered time recurrence algorithm, having
Effect wave number section asks for matching factor, it is contemplated that the pathosis that coefficient is solved, using the conjugate gradient iterative algorithm of pretreatment,
To ask for optimal matching factor, when variable density medium is processed, numerical solidification can be preferably suppressed, while can improve
Group velocity precision.
Further, the conjugate gradient iterative algorithm of the pretreatment utilizes single order sound wave partial differential equations analytic solutions by the time
Difference is unified with space parallax split-phase, and time and space difference operator is carried out in wave-number domain using conjugate gradient method simultaneously excellent
Change, can more efficiently suppress numerical solidification, and improve the computational accuracy of earthquake group velocity to a certain extent.
Further, the matching factor is the time recursion matching factor on the basis of conventional spatial intersecting grid, can be real
Optimize while between current with space difference.
Compared with the technology of the data of existing treatment complicated earth surface, this method mainly has 4 advantages：
(1) on the basis of standard staggered-mesh, it is proposed that a kind of new time recursion matching factor determines method, the recursion
The wave field precision that method is obtained is higher.
(2) single order sound wave partial differential equations analytic solutions are utilized by traditional spatial intersecting grid difference form and precise time
Recurrence algorithm is combined, and makes simulation precision at two aspects of the time and space while being optimized, finally time difference with
Space difference reaches high accuracy simultaneously, and significant effect can be especially reached in the case of exponent number high.
(3) conventional method asks for coefficient using the Taylor expansion near zero, higher in lower wave number range accuracy, with wave number
Increase, error quickly increases, to improve difference scheme of the precision frequently with higher order, this considerably increases amount of calculation.This
Method optimizes time difference and space difference simultaneously in whole Effective wave number section, can be reached identical with less difference order
Precision, so as to improve computational efficiency.
(4) because true underground medium had both had the change that velocity variations there is also density, a host of facts show single order pressure
- rate equation is more favorable for processing variable density medium, therefore this method simulation precision closer to true underground medium.
Brief description of the drawings
Fig. 1 is the wave field snapshot plotting that exponent number is obtained for t=1s moment tradition staggered-mesh difference method simulation in the case of M=4；
Fig. 2 is respectively the wave field snapshot that the difference method simulation after t=1s time optimizations in the case of exponent number is M=4 is obtained
Figure；
Fig. 3 is the Marmousi2 modeling velocity spaces used by test；
Fig. 4 is the Marmousi2 model densities used by test；
Fig. 5 is the single shot record obtained using traditional positive algorithm；
Fig. 6 is the single shot record obtained using forward simulation of the present invention；
Fig. 7 is the partial enlarged drawing of the single shot record obtained using traditional positive algorithm；
Fig. 8 is the single shot record partial enlarged drawing that application invention forward simulation is obtained.
Specific embodiment
To enable above and other objects, features and advantages of the invention to become apparent, it is cited below particularly go out preferred embodiments,
And coordinate institute's accompanying drawings, it is described in detail below.
Embodiment 1.
For numerical value frequently, can start with from optimization difference coefficient, the improvement in terms of three below can be carried out：(1) based on plane
Ripple principle, derives differential operator and filter response of the difference operator in wave-number domain；And make difference using Function Fitting algorithm
Operator's approximation differential operator；(2) space difference is combined with spectral factorization in wave-number domain, improves difference operator and reduce calculating
Error.(3) finite difference formulations precision is improved in terms of grid node.
Staggered-mesh the Forward Modeling based on time-space domain optimization can be achieved by the following technical measures：Step 1, speed
The plane wave solution of-pressure equation group, as a result by asking for accurate staggered time recurrence algorithm coefficient later；Step 2,
Staggered-mesh time recurrence algorithm, target is to approach the equation left side in wave-number domain with the function on the right of equation accurately to solve；Step
Rapid 3, least-squares iteration is solved.
In step 1, it is considered in the case of two-dimentional Chang Midu and constant velocity, obtain single order pressure-speed partial differential equations,
Then ODE is obtained by the utilization space integral transformation in cartesian coordinate system, this ODE equation is done
Time integral obtains plane wave general solution, equation both sides is done with two-dimensional space Fourier transformation and time integral respectively and omits normal
It is several, obtain polarizing the general solution of velocity component.Here, the analytical expression of polarization speed is by the analytic solutions of pressure field
Show, by with the asking for of accurate staggered time recurrence algorithm coefficient later, this is also we to the expression formula
Where the basis of method.
In step 2, staggered-mesh has simulation precision higher compared to conventional regular grid, first according to formula
Accurate staggered time recurrence algorithm is obtained, the time recursion of wave field can be realized by Fourier inversion in formula, while should
The wave field precision that recurrence method is obtained is higher.However, the huge amount of calculation that positive inversefouriertransform is bringing limit its
Application in migration imaging.It is high and take into account higher in view of time-space domain staggered-mesh finite difference simulation computational efficiency
Precision, traditional spatial intersecting grid difference form is combined with precise time recurrence algorithm, makes simulation precision in the time
Optimized simultaneously with two, space aspect.Obtain one kind spatial intersecting difference and realize staggered time recurrence algorithm, then
Time and space difference is carried out into optimization simultaneously.It is final to reach high accuracy simultaneously in time difference and space difference.
Because the analytical expression for polarizing speed is represented by the analytic solutions of pressure field, it is contemplated that time difference and space difference
Both sides content, so the staggered time recurrence algorithm is combined time difference with space difference, refer to by analytic solutions
Bring spatial intersecting grid difference form, the recurrence algorithm for obtaining as staggered time recurrence algorithm into.Spatial intersecting map declination
Cellular uses staggered-mesh finite difference schemes of the prior art.
In step 3, matching factor is changed by objective function so that matching factor is in certain wave-number range
Precise time Recursion Operator is approached as far as possible；Discrete sampling at equal intervals is carried out to object function equation, by asking for for coefficient
Be converted to least squares problem.In view of the pathosis and multi-solution of optimization problem, using the conjugate gradient of pretreatment
Iterative solution method increases the stability of solution.
Application implementation.
(1) choosing flat bed model carries out forward simulation, and upper interval velocity is 1.3km/s, and lower interval velocity is 3.2km/s；On
Layer density is 1.7g/cm3, lower layer density is 2.7g/cm3, sizing grid is 501 × 501, and grid spacing is
Δ x=Δs z=10m.Calculating parameter is：It is the Ricker wavelet of 20Hz as focus to use dominant frequency, and highest frequency is about 60Hz,
The mesh coordinate of focus is (251,201), calculates time step and is set as 1ms.
(2) Fig. 1 and Fig. 2 is respectively the t=1s moment tradition staggered-mesh difference method simulation in the case of exponent number is M=4
The wave field snapshot for obtaining, the wave field snapshot that the difference method simulation after optimization is obtained, conventional method has analog result obvious
Dispersion phenomenon, frequency dispersion is suppressed after optimization；
(3) choosing international standard Marmousi2 models carries out forward simulation test, the horizontal 851 CDP points of the model,
467 sampled points in longitudinal direction, vertically and horizontally sampling interval be 10m, model velocity excursion is 1428m/s to 4700m/s,
As shown in Figure 3；
(4) Fig. 4 show density model corresponding with Fig. 3 rate patterns, and variable density scope is 1.01g/cm3Arrive
2.627g/cm3；
(5) forward simulation calculating parameter is：It is the Ricker wavelet of 20Hz as focus to use dominant frequency, and highest frequency is about
60Hz, calculating time step is Δ t=1ms；Fig. 5 and Fig. 6 are respectively the earthquake record that conventional method is obtained with the present invention,
Two methods use identical difference order M=4, have used PML boundary processing methods.
(6) in order to preferably compare the numerical simulation result of two methods, Fig. 7 and Fig. 8 respectively show conventional method with
The partial enlarged drawing of the earthquake record that the present invention is obtained；
The partial enlarged drawing (Fig. 7 and Fig. 8) of two methods is contrasted, can be clearly seen that conventional method just drills earthquake note
Record has an obvious numerical solidification (in the range of Fig. 7 dotted lines), and the earthquake record that the inventive method is obtained, and numerical solidification is obtained
Preferably compacting is (in the range of Fig. 8 dotted lines).So as to demonstrate advantage of the present invention in terms of numerical solidification is suppressed and to multiple
The adaptability of parasitic mode type.
As can be seen from the above description, the embodiment of the present invention proposes a kind of new time recursion matching factor and determines
Method；Time difference and space parallax split-phase are unified using single order sound wave partial differential equations analytic solutions, and uses conjugation ladder
Degree method carries out optimization simultaneously to time and space difference operator in wave-number domain.The present invention is carried on the basis of standard staggered-mesh
Go out a kind of new time recursion matching factor and determine method.Using single order sound wave partial differential equations analytic solutions by the time difference
Divide and unify with space parallax split-phase, and optimization simultaneously is carried out to time and space difference operator in wave-number domain using conjugate gradient method,
Numerical solidification can be more efficiently suppressed, and improves the computational accuracy of earthquake group velocity to a certain extent.Using it
Although its staggered-mesh the Forward Modeling can also realize same purpose, its precision and computational efficiency are relatively low.
The preferred embodiments of the present invention are the foregoing is only, is not intended to limit the invention, for the technology of this area
For 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. the staggered-mesh the Forward Modeling for being optimized based on time-space domain, it is characterised in that：The method comprises the following steps：
Step 1：Single order pressure-speed partial differential equations form is given, its plane wave general solution is obtained, single order ACOUSTIC WAVE EQUATION is obtained
In pressure field with polarization velocity field analytical expression, wherein, polarize speed analytical expression with the parsing of pressure field
Solution is represented；
Step 2：Spatial intersecting grid difference form is combined with time recurrence algorithm, plane wave general solution is brought into, when obtaining staggeredly
Between recurrence algorithm；
Step 3：Staggered-mesh difference operator is solved using least-squares iteration in wave-number domain；
Step 4：The difference operator that will optimize is used for Acoustic Forward Modeling.
It is 2. as claimed in claim 1 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：In step
In 1, the partial differential equation to single order pressure-speed partial differential equations are integrated by the utilization space in cartesian coordinate system
Conversion obtains ODE；Time integral is done to this ODE equation and obtains plane wave general solution；To equation both sides point
Two-dimensional space Fourier transformation and time integral are not done and constant term is omitted, and obtain polarizing the general solution of velocity component；Here,
The analytical expression for polarizing speed is represented by the analytic solutions of pressure field.
It is 3. as claimed in claim 1 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：In step
In 2, with space difference be combined together time difference by staggered time recurrence algorithm, brings plane wave general solution into, respectively obtains
Time difference and space difference wave-number domain response, will analytic solutions bring spatial intersecting grid difference form into, when obtaining
Between difference and space difference wave-number domain accurate response, target is that in wave-number domain to approach equation with the function on the right of equation left
The accurate solution on side, finally realizes time difference with space difference while reaching high accuracy.
It is 4. as claimed in claim 1 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：In step
In 3, target is to approach the equation left side in wave-number domain with the function on the right of equation accurately to solve, and is changed by objective function
Matching factor so that matching factor approaches precise time Recursion Operator as far as possible in certain wave-number range；To target
Functional equation carries out discrete sampling at equal intervals, asking for coefficient being converted to least squares problem and being iterated solution,
Obtain high accuracy staggered-mesh difference operator.
It is 5. as claimed in claim 1 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：In step
In 3, using time interleaving difference operator wave number domain response as object function, and made using space difference operator wave number domain response
For basic function carries out least square approximation, problem can be summarized as least squares problem, using the conjugate gradient of pretreatment
Iterative solution method increases the stability of solution.
It is 6. as claimed in claim 5 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：In step
In 3, being solved using least-squares iteration is included：By staggered time recurrence algorithm, matching factor is asked in Effective wave number section,
In view of the pathosis that coefficient is solved, using the conjugate gradient iterative algorithm of pretreatment, to ask for optimal matching factor,
When variable density medium is processed, numerical solidification can be preferably suppressed, while group velocity precision can be improved.
It is 7. as claimed in claim 7 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：It is described pre-
The conjugate gradient iterative algorithm for the treatment of is united time difference and space parallax split-phase using single order sound wave partial differential equations analytic solutions
One, and optimization simultaneously is carried out to time and space difference operator in wave-number domain using conjugate gradient method, can be more efficiently
Compacting numerical solidification, and the computational accuracy of earthquake group velocity is improved to a certain extent.
It is 8. as claimed in claim 7 to be based on the staggered-mesh the Forward Modeling that time-space domain optimizes, it is characterised in that：Described
Distribution coefficient is the time recursion matching factor on the basis of conventional spatial intersecting grid, is capable of achieving time and space difference
Optimize simultaneously.
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| CN112818494A (en) * | 2021-02-10 | 2021-05-18 | 西北工业大学 | Functional gradient flow pipe modal and response analysis method based on differential quadrature method |
| CN114444268A (en) * | 2021-12-29 | 2022-05-06 | 中国船舶重工集团公司第七一五研究所 | Underwater sediment layer multi-layer medium acoustic reflection time domain response analytical solution calculation method based on wave group analysis |
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