基于精细积分法的三维弹性波数值模拟(英文)
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摘要
波动方程有限差分法是地震数值模拟中的一种重要的方法,对理解和分析地震传播规律、分析地震属性和解释地震资料有着非常重要的意义。但是有限差分法由于其离散化的思想,产生了不稳定性。精细积分法在有限差分法的基础上,在时间域采用解析解的表达形式,在空间域保留任意差分格式,发展成为半解析的数值方法。本文结合并发展了以往学者的成果,推导了任意精细积分法的三维弹性波正演模拟计算公式,并对其稳定性进行了数值分析。在计算实例中,实现了精细积分法二维和三维弹性波模型的地震正演模拟,对计算结果的分析表明,精细积分法反射信号走时准确,稳定性好,弹性波场相较于声波波场,弹性波波场成分更为丰富,包含了更多波型成分(PP-和PS-反射波、透射波和绕射波),这对实际地震资料的解释和储层分析有重要的意义。实践证明,该方法可直接应用到弹性波的地质模型的数值模拟中。
The Finite Difference (FD) method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data. However, because of its discretization, the FD method is only stable under certain conditions. The Arbitrary Difference Precise Integration (ADPI) method is based on the FD method and adopts an integration scheme in the time domain and an arbitrary difference scheme in the space domain. Therefore, the ADPI method is a semi-analytical method. In this paper, we deduce the formula for the ADPI method based on the 3D elastic equation and improve its stability. In forward modeling cases, the ADPI method was implemented in 2D and 3D elastic wave equation forward modeling. Results show that the travel time of the reflected seismic wave is accurate. Compared with the acoustic wave field, the elastic wave field contains more wave types, including PS- and PP- reflected waves, transmitted waves, and diffracted waves, which is important to interpretation of seismic data. The method can be easily applied to elastic wave equation numerical simulations for geological models.
The Finite Difference (FD) method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data. However, because of its discretization, the FD method is only stable under certain conditions. The Arbitrary Difference Precise Integration (ADPI) method is based on the FD method and adopts an integration scheme in the time domain and an arbitrary difference scheme in the space domain. Therefore, the ADPI method is a semi-analytical method. In this paper, we deduce the formula for the ADPI method based on the 3D elastic equation and improve its stability. In forward modeling cases, the ADPI method was implemented in 2D and 3D elastic wave equation forward modeling. Results show that the travel time of the reflected seismic wave is accurate. Compared with the acoustic wave field, the elastic wave field contains more wave types, including PS- and PP- reflected waves, transmitted waves, and diffracted waves, which is important to interpretation of seismic data. The method can be easily applied to elastic wave equation numerical simulations for geological models.
引文
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Duan,Y.T.,Hu T.Y.,and Li,J.Y.,2011,Application ofa 3-D precise integration method for seismic modelingbased on GPU: 81st Ann.Internat.Mtg.,Soc.of Expl.Geophys.,Expanded Abstract,2850– 2854.
Etgen,J.T.,and O'Brien,M.J.,2007,Computationalmethods for large-scale 3D acoustic finite-differencemodeling: A tutorial: Geophysics,72,SM223– SM230.
Jia,X.F.,Wang,R.Q.,and Hu,T.Y.,2003,The arbitrarydifference precise integration method for solving theseismic wave equation: Earthquake Research in China,19(3),236– 241.
Li,J.Y.,Tang G.Y.,and Hu,T.Y.,2010,Optimization ofa precise integration method for seismic modeling basedon graphic processing unit: Earthquake Science,23,387– 393.
Marfurt,K.J.,1984,Accuracy of finite-difference andfinite-element modeling of the scalar and elastic waveequations: Geophysics,49,533– 549.
Mou,Y.G.,and Pei,Z.L.,2005,Seismic numericalmodeling for 3-D complex media: Petroleum IndustryPress,China,48-58.
Qiang,S.Z.,Wang,X.G.,Tang,M.L.,and Liu,M.,1999,On the arbitrary difference precise integration methodand its numerical stability: Applied Mathematics andMechanics,20(3),256– 262.
Tang,G.Y.,Hu,T.Y.,and Yang,J.H.,2007,Applicationsof a precise integration method in forward seismicmodeling: 77thAnn.Internat.Mtg.,Soc.of Expl.Geophys.,Expanded Abstract,2130– 2133.
Virieux,J.,1986,P-SV wave propagation in heterogeneousmedia: Velocity stress finite difference method: Geophysics,51,889– 901.
Wang,R.Q.,Geng,W.F.,and Wang,S.X.,2003,Seismic forward modeling by arbitrary difference preciseintegration: Oil Geophysical Prospecting,38(3),252–257.
Wang,R.Q.,Zhang,M.,Fei,J.B.,and Huang,W.F.,2006,Precise integral method for 3-D seismic forwardmodeling: Progress in Exploration Geophysics,29(6),394– 397.
Wang,M.F.,and Qu,G.D.,2004,Assessment andimprovement of precise time step integration method:Chinese Journal of Computation Mechanics,21(6),728–733.
Zhong,W.X.,1994,On precise time-integration methodfor structural dynamics: Journal of Dalian University ofTechnology,34(2),131– 136.
Zhong,W.X.,1995,Subdomain precise integration methodand numerical solution of partial differential equations:Computational Structural Mechanics and Applications,12(2),253– 260.
Zhong,W.X.,1996,Single point subdomain preciseintegration and finite difference: Chinese Journal ofTheoretical and Applied Mechanics,28(2),159– 163.
Alford,R.M.,Kelly,K.R.,and Boore,D.M.,1974,Accuracy of finite-difference modeling of the acoustic equation: Geophysics,39,834– 842.
Duan,Y.T.,Hu T.Y.,and Li,J.Y.,2011,Application ofa 3-D precise integration method for seismic modelingbased on GPU: 81st Ann.Internat.Mtg.,Soc.of Expl.Geophys.,Expanded Abstract,2850– 2854.
Etgen,J.T.,and O'Brien,M.J.,2007,Computationalmethods for large-scale 3D acoustic finite-differencemodeling: A tutorial: Geophysics,72,SM223– SM230.
Jia,X.F.,Wang,R.Q.,and Hu,T.Y.,2003,The arbitrarydifference precise integration method for solving theseismic wave equation: Earthquake Research in China,19(3),236– 241.
Li,J.Y.,Tang G.Y.,and Hu,T.Y.,2010,Optimization ofa precise integration method for seismic modeling basedon graphic processing unit: Earthquake Science,23,387– 393.
Marfurt,K.J.,1984,Accuracy of finite-difference andfinite-element modeling of the scalar and elastic waveequations: Geophysics,49,533– 549.
Mou,Y.G.,and Pei,Z.L.,2005,Seismic numericalmodeling for 3-D complex media: Petroleum IndustryPress,China,48-58.
Qiang,S.Z.,Wang,X.G.,Tang,M.L.,and Liu,M.,1999,On the arbitrary difference precise integration methodand its numerical stability: Applied Mathematics andMechanics,20(3),256– 262.
Tang,G.Y.,Hu,T.Y.,and Yang,J.H.,2007,Applicationsof a precise integration method in forward seismicmodeling: 77thAnn.Internat.Mtg.,Soc.of Expl.Geophys.,Expanded Abstract,2130– 2133.
Virieux,J.,1986,P-SV wave propagation in heterogeneousmedia: Velocity stress finite difference method: Geophysics,51,889– 901.
Wang,R.Q.,Geng,W.F.,and Wang,S.X.,2003,Seismic forward modeling by arbitrary difference preciseintegration: Oil Geophysical Prospecting,38(3),252–257.
Wang,R.Q.,Zhang,M.,Fei,J.B.,and Huang,W.F.,2006,Precise integral method for 3-D seismic forwardmodeling: Progress in Exploration Geophysics,29(6),394– 397.
Wang,M.F.,and Qu,G.D.,2004,Assessment andimprovement of precise time step integration method:Chinese Journal of Computation Mechanics,21(6),728–733.
Zhong,W.X.,1994,On precise time-integration methodfor structural dynamics: Journal of Dalian University ofTechnology,34(2),131– 136.
Zhong,W.X.,1995,Subdomain precise integration methodand numerical solution of partial differential equations:Computational Structural Mechanics and Applications,12(2),253– 260.
Zhong,W.X.,1996,Single point subdomain preciseintegration and finite difference: Chinese Journal ofTheoretical and Applied Mechanics,28(2),159– 163.
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