黏弹介质波动方程有限差分解的稳定性研究
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摘要
稳定性问题是地震波数值模拟的一个重要问题.基于地震波传播理论,从黏弹介质本构方程出发,对矩形网格下不同黏弹模型波动方程有限差分解的稳定性进行了理论分析,导出了Kelvin-Voigt黏弹模型和Maxwell黏弹模型在任意空间差分精度下稳定性条件的表达式;给出了品质因子Q≥5时的简化式,并通过数值算例验证了理论研究结论的正确性;总结了地震波速度、频率、空间网格大小、差分系数以及品质因子与稳定性条件的关系;通过误差分析给出了近似公式的使用条件.
The stability problem is a very important aspect in seismic wave numerical simulation.Based on the theory of seismic waves and constitutive relations of visco-elastic models,the stability problem of finite difference scheme with rectangular grids for two visco-elastic models is analyzed.Expressions of stability condition under arbitrary spatial accuracy for Kelvin-Voigt and Maxwell models are derived.With an approximation of medium quality factor Q≥5,a simplified expression is developed and some digital models are given to show the validity of theoretical results.Influence of velocity,frequency,size of grid,differential coefficients,as well as quality factor,on the stability condition is summarized.Finally,the working condition of the simplified stability expression is given by means of its error analysis.
The stability problem is a very important aspect in seismic wave numerical simulation.Based on the theory of seismic waves and constitutive relations of visco-elastic models,the stability problem of finite difference scheme with rectangular grids for two visco-elastic models is analyzed.Expressions of stability condition under arbitrary spatial accuracy for Kelvin-Voigt and Maxwell models are derived.With an approximation of medium quality factor Q≥5,a simplified expression is developed and some digital models are given to show the validity of theoretical results.Influence of velocity,frequency,size of grid,differential coefficients,as well as quality factor,on the stability condition is summarized.Finally,the working condition of the simplified stability expression is given by means of its error analysis.
引文
董良国,马在田,曹景忠.2000.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,43(6):856-864.
牟永光,裴正林.2005.三维复杂介质地震数值模拟[M].北京:石油工业出版社:33-34.
孙成禹.2007.地震波理论与方法[M].东营:中国石油大学出版社:219-221.
孙成禹,印兴耀.2007.三参数常Q黏弹性模型构造方法研究[J].地震学报,29(4):348-357.
Aki K,Richards P G.1980.Quantitative Seismology:Theoryand Method[M].New York:W HFreeman and Compa-ny:167-187.
Carcione J M,Herman G C,ten Kroode A P E.2002.Seismic modeling[J].Geophysics,67(4):1304-1325.
de Basabe J D,Sen K.2007.Grid dispersion and stability criteria of some common finite difference and finite elementmethods for acoustic and elastic wave propagation[C/CD]∥Expanded Abstracts of SEG77th Annual Meeting:1992-1996.
Hesthoim S.2003.Elastic wave modeling withfree surfaces:Stability of long simulations[J].Geophysics,68(1):314-321.
Ilan A,Loewenthal D.1976.Instability of finite-difference schemes due to boundary conditionsin elastic media[J].Geo-physical Prospecting,24:431-453.
Levander A R.1988.Fourth-order finite-difference P-SVseismograms[J].Geophysics,53(11):1425-1436.
Manning P M,Margrave G F.2000.Finite difference modeling analysis,dispersion and stability[C/CD]∥Expanded Ab-stracts of SEG70th Annual Meeting.Calgary,Canada:1-4.
Moczo P,Kristek J,Bystricky E.2000.Stability and grid dispersion of the P-SV4th-order staggered-grid finite-differ-ence schemes[J].Geophysics,44:381-402.
Robertsson J O A,Blanch J O,Symes W W.1994.Viscoelastic finite-difference modeling[J].Geophysics,59(9):1444-1456.
Saenger E H,Bohlen T.2004.Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rota-ted staggered grid[J].Geophysics,69(2):583-591.
Vidale J E,Clayton R W.1986.A stable free-surface boundary condition for two-dimensional elastic finite-differencewave simulation[J].Geophysics,51(12):2247-2249.
Virieux J.1986.P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method[J].Geophy-sics,51(4):889-901.
牟永光,裴正林.2005.三维复杂介质地震数值模拟[M].北京:石油工业出版社:33-34.
孙成禹.2007.地震波理论与方法[M].东营:中国石油大学出版社:219-221.
孙成禹,印兴耀.2007.三参数常Q黏弹性模型构造方法研究[J].地震学报,29(4):348-357.
Aki K,Richards P G.1980.Quantitative Seismology:Theoryand Method[M].New York:W HFreeman and Compa-ny:167-187.
Carcione J M,Herman G C,ten Kroode A P E.2002.Seismic modeling[J].Geophysics,67(4):1304-1325.
de Basabe J D,Sen K.2007.Grid dispersion and stability criteria of some common finite difference and finite elementmethods for acoustic and elastic wave propagation[C/CD]∥Expanded Abstracts of SEG77th Annual Meeting:1992-1996.
Hesthoim S.2003.Elastic wave modeling withfree surfaces:Stability of long simulations[J].Geophysics,68(1):314-321.
Ilan A,Loewenthal D.1976.Instability of finite-difference schemes due to boundary conditionsin elastic media[J].Geo-physical Prospecting,24:431-453.
Levander A R.1988.Fourth-order finite-difference P-SVseismograms[J].Geophysics,53(11):1425-1436.
Manning P M,Margrave G F.2000.Finite difference modeling analysis,dispersion and stability[C/CD]∥Expanded Ab-stracts of SEG70th Annual Meeting.Calgary,Canada:1-4.
Moczo P,Kristek J,Bystricky E.2000.Stability and grid dispersion of the P-SV4th-order staggered-grid finite-differ-ence schemes[J].Geophysics,44:381-402.
Robertsson J O A,Blanch J O,Symes W W.1994.Viscoelastic finite-difference modeling[J].Geophysics,59(9):1444-1456.
Saenger E H,Bohlen T.2004.Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rota-ted staggered grid[J].Geophysics,69(2):583-591.
Vidale J E,Clayton R W.1986.A stable free-surface boundary condition for two-dimensional elastic finite-differencewave simulation[J].Geophysics,51(12):2247-2249.
Virieux J.1986.P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method[J].Geophy-sics,51(4):889-901.
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