二维地震波场有限差分法数值模拟研究
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摘要
地震波场数值模拟是勘探地震学的重要研究课题之一,也是研究波动现象的重要手段,在油气田勘探开发中发挥着重要的作用。地震波场数值模拟常用的方法有伪谱法、有限差分法、有限元法等。有限差分法具有计算速度快、占用内存小等优点,该方法对于近远场及复杂边界都有广泛的适用性,能够准确地模拟波在各种介质及复杂结构地层中的传播规律,是勘探地震学中应用最广泛的数值计算方法。
本文以波动理论为理论基础,利用泰勒级数展开式推导出波动方程的有限差分格式及其离散表达式。针对传统二阶精度差分方法模拟精度较低的问题,推导出时间域二阶空间域四阶精度的有限差分方程,并综合分析了初始条件、震源项、算法稳定性及数值频散等因素在有限差分法数值模拟中的影响。
有限差分数值模拟计算是在一个有限的区域内进行的,当波传播到边界时就会产生边界反射,这是进行数值模拟计算时所不期望出现的。论文通过调研大量文献资料,对常用的边界条件进行归纳对比,并对本文选用的透明边界条件和吸收边界条件两种方法进行了详细分析。在此基础上,分别推导得到这两种边界条件方法的离散表达式,并在数值模拟过程中进行验证,结果表明这两种边界条件对边界反射都有较好的吸收效果。
通过建立均匀模型、层状模型及背斜向斜模型等各种理论模型,基于FORTRAN语言编程,论文实现了波动方程有限差分算法的数值模拟。对不同地质理论模型进行模拟,从模拟结果的对比分析中可以看出,模型参数的选择及各参数间的相互关系对模拟结果有着显著影响。无论是模拟精度,还是模拟计算效率,有限差分算法都具有一定优势。
通过综合研究,论文认为波动方程有限差分算法具有算法简单、计算效率高、模拟精度较高等特点,理论研究意义大,应用前景广阔。
Numerical simulation of seismic wave is one of the important research topics of exploration seismology, it is also an important means of volatility, in the oil and gas exploration and development it plays an important role. Numerical simulation of seismic wave field methods commonly used pseudo-spectral method, finite difference method, finite element method. Finite difference method has advantages in it’s calculation speed and small memory, the method has broad applicability in the near field and the complexity of the border, and it can accurately simulate wave propagation in various media and the formation of complex structures in the spread of laws, the method is the most widely used numerical method in exploration seismology.
In this paper, the theoretical basis for the wave theory, the use of Taylor series expansion of the wave equation finite-difference equations and discrete expressions are derived in detail, for the traditional second-order accuracy finite difference method simulation of the problem of low precision, derived from time-domain second-order spatial accuracy of fourth-order finite-difference equations. Paper also comprehensively analyzed the impact of the initial conditions, the source, the algorithm stability and numerical dispersion factors in the numerical simulation of the finite difference method.
Finite difference numerical simulation is limited in a region, when the wave propagation to the border, the border will result in reflection, which is not expect to arise. Through a large number of research literature research, the paper summarized and compared some common boundary conditions, then analyzed the transparent selection of boundary conditions and absorbing conditions in detail. On this basis, deduced and received discrete expression of the two boundary conditions. Numerical simulation in the verification process shows that these two kinds of boundary conditions on the border have a good reflection of the absorption.
Through the establishment of uniform model, layered model and anticline syncline model various theoretical models, based on the FORTRAN programming language, paper achieved the numerical simulation of finite difference algorithm. To simulate different geological theoretical models, from the comparative analysis of the simulation results we can see that ,the choice of model parameters and the relationship between the parameters have a significant impact on the simulation results. Finite difference algorithm has certain advantages in both the simulation accuracy and the efficiency of simulation.
Through integrated research, thesis that the wave equation finite-difference algorithm is simple, computationally efficient, simulation of the characteristics of high accuracy, it has a great meaning of theoretical research and broad application prospects.
本文以波动理论为理论基础,利用泰勒级数展开式推导出波动方程的有限差分格式及其离散表达式。针对传统二阶精度差分方法模拟精度较低的问题,推导出时间域二阶空间域四阶精度的有限差分方程,并综合分析了初始条件、震源项、算法稳定性及数值频散等因素在有限差分法数值模拟中的影响。
有限差分数值模拟计算是在一个有限的区域内进行的,当波传播到边界时就会产生边界反射,这是进行数值模拟计算时所不期望出现的。论文通过调研大量文献资料,对常用的边界条件进行归纳对比,并对本文选用的透明边界条件和吸收边界条件两种方法进行了详细分析。在此基础上,分别推导得到这两种边界条件方法的离散表达式,并在数值模拟过程中进行验证,结果表明这两种边界条件对边界反射都有较好的吸收效果。
通过建立均匀模型、层状模型及背斜向斜模型等各种理论模型,基于FORTRAN语言编程,论文实现了波动方程有限差分算法的数值模拟。对不同地质理论模型进行模拟,从模拟结果的对比分析中可以看出,模型参数的选择及各参数间的相互关系对模拟结果有着显著影响。无论是模拟精度,还是模拟计算效率,有限差分算法都具有一定优势。
通过综合研究,论文认为波动方程有限差分算法具有算法简单、计算效率高、模拟精度较高等特点,理论研究意义大,应用前景广阔。
Numerical simulation of seismic wave is one of the important research topics of exploration seismology, it is also an important means of volatility, in the oil and gas exploration and development it plays an important role. Numerical simulation of seismic wave field methods commonly used pseudo-spectral method, finite difference method, finite element method. Finite difference method has advantages in it’s calculation speed and small memory, the method has broad applicability in the near field and the complexity of the border, and it can accurately simulate wave propagation in various media and the formation of complex structures in the spread of laws, the method is the most widely used numerical method in exploration seismology.
In this paper, the theoretical basis for the wave theory, the use of Taylor series expansion of the wave equation finite-difference equations and discrete expressions are derived in detail, for the traditional second-order accuracy finite difference method simulation of the problem of low precision, derived from time-domain second-order spatial accuracy of fourth-order finite-difference equations. Paper also comprehensively analyzed the impact of the initial conditions, the source, the algorithm stability and numerical dispersion factors in the numerical simulation of the finite difference method.
Finite difference numerical simulation is limited in a region, when the wave propagation to the border, the border will result in reflection, which is not expect to arise. Through a large number of research literature research, the paper summarized and compared some common boundary conditions, then analyzed the transparent selection of boundary conditions and absorbing conditions in detail. On this basis, deduced and received discrete expression of the two boundary conditions. Numerical simulation in the verification process shows that these two kinds of boundary conditions on the border have a good reflection of the absorption.
Through the establishment of uniform model, layered model and anticline syncline model various theoretical models, based on the FORTRAN programming language, paper achieved the numerical simulation of finite difference algorithm. To simulate different geological theoretical models, from the comparative analysis of the simulation results we can see that ,the choice of model parameters and the relationship between the parameters have a significant impact on the simulation results. Finite difference algorithm has certain advantages in both the simulation accuracy and the efficiency of simulation.
Through integrated research, thesis that the wave equation finite-difference algorithm is simple, computationally efficient, simulation of the characteristics of high accuracy, it has a great meaning of theoretical research and broad application prospects.
引文
[1]裴正林,牟永光.地震波传播数值模拟.地球物理学进展[J],2004,19(4):933~941.
[2]马在田.计算地球物理学概论[M].上海:同济大学出版社,1997.
[3]何樵登.地震波理论[M] .地质出版社,1988
[4]何樵登,熊维钢.应用地球物理教程——地震勘探[M] .地质出版社,1991
[5]吴清岭,张平,施泽龙.波动方程正演模拟即应用[J].大庆石油地质与开发,1998,17(3)
[6]江玉乐,雷苑著.地球物理数据处理教程[M].北京:地质大学出版社,2006,7
[7]姚姚著.地震波场与地震勘探.北京:地质出版社,2006,6.
[8]张永刚.地震波数值模拟方法[J].石油物探,2003,42(2):143-147.
[9]陈伟.起伏地表条件下二维地震波场的数值模拟[J].勘探地球物理进展,2005,28(1)
[30]JOSCM. CARCIONE, GCRARD C. HERMAN, et al.Y2K Review Article: Seismic modeling [J]. Geophysics, 2002, 67 (4):1304.
[10]吴永刚,吴清岭.有限差分法弹性波场数值模拟[J].大庆石油地质与开发,1994,13(3)
[11]Alterman. Propagation of elastic wave in layered media by finite difference methods. Bulletin of the Seismological Society of America,1968;58(1):367~398
[12]Alford, R. M. Kelly ,K .R. and Booer, D. M. Accuracy of finite difference modeling of the acoustic wave equation.Geophysics,1974;39(6):834~842
[13]周家纪,贺振华.模拟地震波传播的大网格快速差分算法[J].地球物理学报,1994;37(增刊):450~454
[14]董良国.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,2000,43(6):856
[15]邵秀民,蓝志凌.非均匀各向同性介质中地震波传播的数值模拟[J].地球物理学报,1935;38(1):39～55
[16]刘洋,李承楚,牟永光.任意偶数阶精度有限差分数值模拟[J].石油地球物理勘探,1998,53(1)
[17]褚春雷,王修田.非规则三角网格有限差分法地震正演模拟[J].中国海洋大学学报,2005,35(1):043～048
[18]董良国,李培明.地震波传播数值模拟中的频散问题[J].天然气工业,2004,24(6):53-56
[19]奚先,姚姚.二维随机介质及波动方程正演模拟[J].石油地球物理勘探,2001,36(5):546～552
[20]董良国等.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
[21]奚先,姚姚.二维弹性随机介质中的波场特征[J].石油地球物理勘探,2004,39(6):679～685
[22]邵秀民,蓝志凌.非均匀各向同性介质中地震波传播的数值模拟[J].地球物理学报,1935;38(1):39～55
[23]孙若昧.地震波传播有限差分模拟的人工边界问题[J].地球物理学进展,1996,11(3):53
[24] Cerjan C, Kosloff D, Kosloff R and ReshefM. A nonreflecting boundary condit -ion for discrete acoustic and elastic wave equation. Geophysics, 1985, 50 (4) : 705～708
[25]Clayton R, Engquist B. Absorbing boundary conditions for acoustic and elastic wave equation. Bulletin of the Seismological Society of America, 1977;66(11):1529~1540
[26]Reynolds. Boundary conditions for the numerical solution of wave propagation problem. Geophysics, 1978;43(6):1099~1110
[27]崔兴福,张关泉.地震波方程人工边界的两种处理方法[J].石油物探,2003,42(4):452～455
[28]张毅.声波正演模拟中的人工边界问题[J].工程地球物理学报,2007,4(4).
[29]李文杰,魏修成,刘洋.声波正演中一种新的边界条件-双重吸收边界条件[J].石油物探,2004,43(6):528～531
[30]邢丽.地震波数值模拟中的吸收边界条件[J].上海第二工业大学学报,2006,23(4).
[31]Dahlain M A. The application of high difference to the scalar waveequation. Geophysics, 1986; 51(1): 54~66
[32] Higdon R L. Absorbing boundary condition for elastic waves. Geophysics,1991, 56 (2) : 231～241
[33]孙成禹,张吉辉.完全纵波方程有限差分数值模拟[J].石油地球物理勘探,2005;40(3):289～294
[34]黄自萍,徐伶俐,周继顺.起伏地表声波方程的数值模拟[J].石油地球物理勘探,2006,41(3):275～280
[35]宛书金,董敏煌等.横向各向同性介质中的地震波传播特性[J].石油大学学报,2002,26(1):23~28
[36]Yu Zhiyuan. A simple and effective method for the reflection extraction in rectangular waveguide discontinuity analysis by the FDTD Microwave and Optical Technology Letters,1997,15(1):57~59.
[38] CollinoF and Tsogka C. Application of the perfectly matched adsorbing layer model to the linear elasto dynamic problem in anisotropic heterogeneous media. Geophysics, 2001, 66 (1) : 294～307
[38] Kelly : synthetic Seismograms—A finite difference approach , Geophysics ,1976 Vol141 , 2～27 .
[39]Tam C,Auriault L,Cambuliy F. Perfectly matched layer as an absorbing boundary condition for the linearized Euler equations in open and ducted domains[J].Journal of Computational Physics,1998,144:213~234
[40] ALBERT : Boundary conditions for the numerical solution of wave propagation problems , Geophysics , 1978 Vol. 43 , 1099～1110
[41]Jung Kyung-Yuung, Kin Hyeongdong, Ko Kwang Cheol. An improved unimodal absorbing boundary condition for waveguide problems. IEEE Microwave and Guided Wave Letters,1997,7(11):368~370.
[42]Hicks G J. Arbitrary source and receiver positioning in finite-difference schemes using Kaiser windowed sinc functions[J].Geophysics,2002,67(1)
[2]马在田.计算地球物理学概论[M].上海:同济大学出版社,1997.
[3]何樵登.地震波理论[M] .地质出版社,1988
[4]何樵登,熊维钢.应用地球物理教程——地震勘探[M] .地质出版社,1991
[5]吴清岭,张平,施泽龙.波动方程正演模拟即应用[J].大庆石油地质与开发,1998,17(3)
[6]江玉乐,雷苑著.地球物理数据处理教程[M].北京:地质大学出版社,2006,7
[7]姚姚著.地震波场与地震勘探.北京:地质出版社,2006,6.
[8]张永刚.地震波数值模拟方法[J].石油物探,2003,42(2):143-147.
[9]陈伟.起伏地表条件下二维地震波场的数值模拟[J].勘探地球物理进展,2005,28(1)
[30]JOSCM. CARCIONE, GCRARD C. HERMAN, et al.Y2K Review Article: Seismic modeling [J]. Geophysics, 2002, 67 (4):1304.
[10]吴永刚,吴清岭.有限差分法弹性波场数值模拟[J].大庆石油地质与开发,1994,13(3)
[11]Alterman. Propagation of elastic wave in layered media by finite difference methods. Bulletin of the Seismological Society of America,1968;58(1):367~398
[12]Alford, R. M. Kelly ,K .R. and Booer, D. M. Accuracy of finite difference modeling of the acoustic wave equation.Geophysics,1974;39(6):834~842
[13]周家纪,贺振华.模拟地震波传播的大网格快速差分算法[J].地球物理学报,1994;37(增刊):450~454
[14]董良国.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,2000,43(6):856
[15]邵秀民,蓝志凌.非均匀各向同性介质中地震波传播的数值模拟[J].地球物理学报,1935;38(1):39～55
[16]刘洋,李承楚,牟永光.任意偶数阶精度有限差分数值模拟[J].石油地球物理勘探,1998,53(1)
[17]褚春雷,王修田.非规则三角网格有限差分法地震正演模拟[J].中国海洋大学学报,2005,35(1):043～048
[18]董良国,李培明.地震波传播数值模拟中的频散问题[J].天然气工业,2004,24(6):53-56
[19]奚先,姚姚.二维随机介质及波动方程正演模拟[J].石油地球物理勘探,2001,36(5):546～552
[20]董良国等.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
[21]奚先,姚姚.二维弹性随机介质中的波场特征[J].石油地球物理勘探,2004,39(6):679～685
[22]邵秀民,蓝志凌.非均匀各向同性介质中地震波传播的数值模拟[J].地球物理学报,1935;38(1):39～55
[23]孙若昧.地震波传播有限差分模拟的人工边界问题[J].地球物理学进展,1996,11(3):53
[24] Cerjan C, Kosloff D, Kosloff R and ReshefM. A nonreflecting boundary condit -ion for discrete acoustic and elastic wave equation. Geophysics, 1985, 50 (4) : 705～708
[25]Clayton R, Engquist B. Absorbing boundary conditions for acoustic and elastic wave equation. Bulletin of the Seismological Society of America, 1977;66(11):1529~1540
[26]Reynolds. Boundary conditions for the numerical solution of wave propagation problem. Geophysics, 1978;43(6):1099~1110
[27]崔兴福,张关泉.地震波方程人工边界的两种处理方法[J].石油物探,2003,42(4):452～455
[28]张毅.声波正演模拟中的人工边界问题[J].工程地球物理学报,2007,4(4).
[29]李文杰,魏修成,刘洋.声波正演中一种新的边界条件-双重吸收边界条件[J].石油物探,2004,43(6):528～531
[30]邢丽.地震波数值模拟中的吸收边界条件[J].上海第二工业大学学报,2006,23(4).
[31]Dahlain M A. The application of high difference to the scalar waveequation. Geophysics, 1986; 51(1): 54~66
[32] Higdon R L. Absorbing boundary condition for elastic waves. Geophysics,1991, 56 (2) : 231～241
[33]孙成禹,张吉辉.完全纵波方程有限差分数值模拟[J].石油地球物理勘探,2005;40(3):289～294
[34]黄自萍,徐伶俐,周继顺.起伏地表声波方程的数值模拟[J].石油地球物理勘探,2006,41(3):275～280
[35]宛书金,董敏煌等.横向各向同性介质中的地震波传播特性[J].石油大学学报,2002,26(1):23~28
[36]Yu Zhiyuan. A simple and effective method for the reflection extraction in rectangular waveguide discontinuity analysis by the FDTD Microwave and Optical Technology Letters,1997,15(1):57~59.
[38] CollinoF and Tsogka C. Application of the perfectly matched adsorbing layer model to the linear elasto dynamic problem in anisotropic heterogeneous media. Geophysics, 2001, 66 (1) : 294～307
[38] Kelly : synthetic Seismograms—A finite difference approach , Geophysics ,1976 Vol141 , 2～27 .
[39]Tam C,Auriault L,Cambuliy F. Perfectly matched layer as an absorbing boundary condition for the linearized Euler equations in open and ducted domains[J].Journal of Computational Physics,1998,144:213~234
[40] ALBERT : Boundary conditions for the numerical solution of wave propagation problems , Geophysics , 1978 Vol. 43 , 1099～1110
[41]Jung Kyung-Yuung, Kin Hyeongdong, Ko Kwang Cheol. An improved unimodal absorbing boundary condition for waveguide problems. IEEE Microwave and Guided Wave Letters,1997,7(11):368~370.
[42]Hicks G J. Arbitrary source and receiver positioning in finite-difference schemes using Kaiser windowed sinc functions[J].Geophysics,2002,67(1)
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