波场模拟中的数值频散分析与校正策略
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摘要
波动方程有限差分法正演模拟,对认识地震波传播规律、进行地震属性研究、地震资料地质解释、储层评价等,均具有重要的理论和实际意义.但有限差分法本身固有存在着数值频散问题,数值频散在正演模拟中是一种严重的干扰,会降低波场模拟的精度与分辨率.针对TI介质波场模拟的交错网格有限差分方法,本文从空间网格离散、时间网格离散和算子近似等三个方面对其产生的数值频散进行了分析,并结合其他学者的研究成果给出了TI介质波场模拟中压制数值频散的方法与策略:在已知介质频散关系时,对差分算子可实施算子校正;通过提高差分方程的阶数来提高波场模拟精度;采用流体力学中守恒式方程的通量校正传输方法来压制波场模拟中的数值频散;在实际正演模拟时,采用交错网格高阶有限差分方程,不仅在空间上采用高阶差分,而且在时间上也要采用高阶差分,否则只在单一方向上(空间或时间)提高方程的阶数对压制数值频散也不会取得理想的效果.
It is well known that finite-difference simulation of elastic wave equation is very important for us to recognize the regular pattern in propagation of seismic wave、perform research of seismic attributes、implement geological interpretation of seismic data and develop the resources of the earth. Although they have been widely implemented for wave extrapolation in modeling and migration, finite-difference schemes for numerically solving the wave equation suffer from undesirable ripples, so-called numerical dispersion. The numerical dispersion interferes with the seismic modeling seriously and decreases the precision of simulation. Here, we discuss what causes numerical dispersion in simulation of elastic wave in TI media according to high-order finite-difference method of staggered grid and present method and strategy to suppress finite-difference errors. For eliminating the numerical dispersion of simulation, we can use high-order finite-difference equation instead of low-order finite-difference equation to improve the precision of simulation; through using the dispersion relation of TI media, finite-difference operators are implemented with correction; the flux-corrected transport (FCT) method, commonly used in hydrodynamics for conservative equations, can then be used in solving elastic wave equations of TI media; for avoiding both space-grid dispersion and time-grid dispersion, is high-order finite-difference operator applied not only to space differentials but also to time differentials.
It is well known that finite-difference simulation of elastic wave equation is very important for us to recognize the regular pattern in propagation of seismic wave、perform research of seismic attributes、implement geological interpretation of seismic data and develop the resources of the earth. Although they have been widely implemented for wave extrapolation in modeling and migration, finite-difference schemes for numerically solving the wave equation suffer from undesirable ripples, so-called numerical dispersion. The numerical dispersion interferes with the seismic modeling seriously and decreases the precision of simulation. Here, we discuss what causes numerical dispersion in simulation of elastic wave in TI media according to high-order finite-difference method of staggered grid and present method and strategy to suppress finite-difference errors. For eliminating the numerical dispersion of simulation, we can use high-order finite-difference equation instead of low-order finite-difference equation to improve the precision of simulation; through using the dispersion relation of TI media, finite-difference operators are implemented with correction; the flux-corrected transport (FCT) method, commonly used in hydrodynamics for conservative equations, can then be used in solving elastic wave equations of TI media; for avoiding both space-grid dispersion and time-grid dispersion, is high-order finite-difference operator applied not only to space differentials but also to time differentials.
引文
[1] 董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):411~419.
[2] 杨慧珠,张友生,陶果.井眼条件下弹性波传播问题的三维有限差分数值模拟[J].地球物理学进展,2003,12(1):348~352.
[3] 孙卫涛,杨慧珠.各向异性介质弹性波传播的三维不规则网格有限差分方法[J].地球物理学报,2004,18(2):332~337.
[4] 杨顶辉,滕吉文.各向异性介质中三分量地震记录的FCT有限差分模拟[J].石油地球物理勘探,1997,32(2):181~191.
[5] 潘冬明,杨顶辉,张慧.含煤层地质环境下地震波场的数值模拟[J].地球物理学进展,1997,12(1):74~83.
[6] HokstadK.3D elastic finite difference modeling in tiltedtransversely isotropic media[J].72thAnn.Internat.Mtg.SocExplGeophysExpandedAbstracts,2002,1951~1954.
[7] FeiT,LarnerK.Elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport[J].Geophysics,1995,60(6):1830~1842.
[8] LiZ.Compensating finite difference errors in3D migrationand modeling[J].Geophysics,1995,56(10):1650~1660.
[9] WintersteinD F.Velocity anisotropy terminology for geophys icists[J].Geophysics,1990,55(8):1070~1088.
[10] LevanderA R.Four order finite differenceP SV seismo grams[J].Geophysics,1988,53(11):1425~1436.
[11] VirieuxJ.P SV wave propagation in heterogeneous media:velocity stress finite difference method[J ].Geophysics,1986,51(40):889~901.
[12] VirieuxJ.SH wave propagation in heterogeneous media: ve locity stress finite difference method[J].Geophysics,1984,49(11):1933~1957.
[13] BorisJ P,BookD L.Flux corrected transport.I.SHAS TA,A fluid transport algorithm that works[J].J ComputPhys,1973,11:38~69.
[14] BookD L,BorisJ P,HainK.Flux corrected transportII:generalization of the method[J].J ComputPhys,1975,18:248~283.
[2] 杨慧珠,张友生,陶果.井眼条件下弹性波传播问题的三维有限差分数值模拟[J].地球物理学进展,2003,12(1):348~352.
[3] 孙卫涛,杨慧珠.各向异性介质弹性波传播的三维不规则网格有限差分方法[J].地球物理学报,2004,18(2):332~337.
[4] 杨顶辉,滕吉文.各向异性介质中三分量地震记录的FCT有限差分模拟[J].石油地球物理勘探,1997,32(2):181~191.
[5] 潘冬明,杨顶辉,张慧.含煤层地质环境下地震波场的数值模拟[J].地球物理学进展,1997,12(1):74~83.
[6] HokstadK.3D elastic finite difference modeling in tiltedtransversely isotropic media[J].72thAnn.Internat.Mtg.SocExplGeophysExpandedAbstracts,2002,1951~1954.
[7] FeiT,LarnerK.Elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport[J].Geophysics,1995,60(6):1830~1842.
[8] LiZ.Compensating finite difference errors in3D migrationand modeling[J].Geophysics,1995,56(10):1650~1660.
[9] WintersteinD F.Velocity anisotropy terminology for geophys icists[J].Geophysics,1990,55(8):1070~1088.
[10] LevanderA R.Four order finite differenceP SV seismo grams[J].Geophysics,1988,53(11):1425~1436.
[11] VirieuxJ.P SV wave propagation in heterogeneous media:velocity stress finite difference method[J ].Geophysics,1986,51(40):889~901.
[12] VirieuxJ.SH wave propagation in heterogeneous media: ve locity stress finite difference method[J].Geophysics,1984,49(11):1933~1957.
[13] BorisJ P,BookD L.Flux corrected transport.I.SHAS TA,A fluid transport algorithm that works[J].J ComputPhys,1973,11:38~69.
[14] BookD L,BorisJ P,HainK.Flux corrected transportII:generalization of the method[J].J ComputPhys,1975,18:248~283.
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