交错网格高阶有限差分法的弹性波波场数值模拟
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摘要
波场数值模拟是研究地球介质中地震波传播的运动学和动力学特征的重要手段,也是进行地震数据处理的基础。地震波正演模拟方法的优劣会直接影响模拟的效果、成像精度、实用化程度以及对速度剧烈变化区域的适应能力等,其核心技术是求解波动方程,模拟地震波在地下介质中的传播规律。为了得到更好的模拟效果,在模拟的过程中,必须考虑人为边界反射以及稳定性、频散等问题。
目前全球能源紧张,地下资源的勘探成为焦点。我国地势地形的复杂性,对起伏地表及夹速层等复杂介质的研究显得尤为重要,因此要求有更客观、更精确的方法来模拟地下波场的传播规律,以达到对地质勘探的目的。
交错网格高阶有限差分是用交错网格取代一般矩形差分网格,通过交错网格的半程计算,可以得到足够高阶的空间和时间精度的差分格式。本文首先从泰勒级数展开式出发,采用交错网格的高阶有限差分法、基于特征值分析的吸收边界法,通过分析稳定性、频散等要求,得到适合弹性波波动方程的任意偶数阶精度的差分格式、各边界及角点处的吸收边界条件差分格式,实现二维弹性波在各种复杂地质模型中的交错网格差分数值模拟。
本文主要研究了水平地表、起伏地表情况下层状介质,包含有断层、夹层等模型的数值模拟,考虑该算法的可行性及优劣性。实验结果表明,交错网格高阶有限差分法能够完成对各种复杂介质的弹性波的波场模拟,显著削弱数值频散,具有精度高、通用性强、稳定性好等特点,能够有效地吸收人为边界反射,可以成功模拟波的传播过程。另外,实验验证了在保证一定精度的前提下,采用较大的空间网格间距,可以提高计算效率。
Wave field numerical simulation is used frequently that is the important method of researching kinematics on seismic wave propagation and dynamic characteristics in earth medium, that is the foundation of seismic data processing. The quality of seismic wave forward modeling methods will directly influence the effect of simulation, precision of migration, practicability and adaptive capacity to the area of rapid velocity change and so on. Its core technology is solving wave equation to simulate the propagation of seismic wave in underground medium. To get the better result of seismic wave propagation, the artificial boundary reflections, stability, numerical dispersion and other problems must be considered in the numerical simulation.
With the global energy scarcity, the exploration of underground resource is pushed into the spotlight. Because of the complexity of topography in China, the research of irregular surface and low-velocity layer and other complex structure is of grate important. Therefore the more objective and more accurate method is required to simulate seismic wave propagation and achieve the goal of geological exploration.
Staggered grid higher order finite difference can substitute staggered grid for the general rectangle difference grid, obtain the difference finite of sufficient higher order spatial precision and timer resolution by means of half grid calculating. Firstly, the Taylor series was expanded and the staggered grid finite difference, absorbing boundary method based on the Eigen value analysis were used to get the difference format of equations for elastic waves and every boundary and angular point by analysis the request of stability and dispersion. This format could obtain arbitrary accuracy to implement the 2-D elastic wave numerical simulation with the staggered grid finite difference in various kinds of complex geologic model.
This paper mainly studied the numerical simulation on horizontal earth surface, irregular surface layered media which have abruption, interlayer and other complex models, thought of the algorithm of feasibility and quality. The numerical experiment results indicate that this staggered grid finite difference could complete the elastic wave field simulation in various kinds of complex geologic model, weakened numerical dispersion markedly, had the characters of high simulation precision, strong commonality good stability and so on, could absorbing artificial boundary reflections effectively, attained good simulation effects of wave propagation. In addition, experiment had proved under the premise of promising precision, bigger mesh spacing could be used to improve calculative efficiency.
目前全球能源紧张,地下资源的勘探成为焦点。我国地势地形的复杂性,对起伏地表及夹速层等复杂介质的研究显得尤为重要,因此要求有更客观、更精确的方法来模拟地下波场的传播规律,以达到对地质勘探的目的。
交错网格高阶有限差分是用交错网格取代一般矩形差分网格,通过交错网格的半程计算,可以得到足够高阶的空间和时间精度的差分格式。本文首先从泰勒级数展开式出发,采用交错网格的高阶有限差分法、基于特征值分析的吸收边界法,通过分析稳定性、频散等要求,得到适合弹性波波动方程的任意偶数阶精度的差分格式、各边界及角点处的吸收边界条件差分格式,实现二维弹性波在各种复杂地质模型中的交错网格差分数值模拟。
本文主要研究了水平地表、起伏地表情况下层状介质,包含有断层、夹层等模型的数值模拟,考虑该算法的可行性及优劣性。实验结果表明,交错网格高阶有限差分法能够完成对各种复杂介质的弹性波的波场模拟,显著削弱数值频散,具有精度高、通用性强、稳定性好等特点,能够有效地吸收人为边界反射,可以成功模拟波的传播过程。另外,实验验证了在保证一定精度的前提下,采用较大的空间网格间距,可以提高计算效率。
Wave field numerical simulation is used frequently that is the important method of researching kinematics on seismic wave propagation and dynamic characteristics in earth medium, that is the foundation of seismic data processing. The quality of seismic wave forward modeling methods will directly influence the effect of simulation, precision of migration, practicability and adaptive capacity to the area of rapid velocity change and so on. Its core technology is solving wave equation to simulate the propagation of seismic wave in underground medium. To get the better result of seismic wave propagation, the artificial boundary reflections, stability, numerical dispersion and other problems must be considered in the numerical simulation.
With the global energy scarcity, the exploration of underground resource is pushed into the spotlight. Because of the complexity of topography in China, the research of irregular surface and low-velocity layer and other complex structure is of grate important. Therefore the more objective and more accurate method is required to simulate seismic wave propagation and achieve the goal of geological exploration.
Staggered grid higher order finite difference can substitute staggered grid for the general rectangle difference grid, obtain the difference finite of sufficient higher order spatial precision and timer resolution by means of half grid calculating. Firstly, the Taylor series was expanded and the staggered grid finite difference, absorbing boundary method based on the Eigen value analysis were used to get the difference format of equations for elastic waves and every boundary and angular point by analysis the request of stability and dispersion. This format could obtain arbitrary accuracy to implement the 2-D elastic wave numerical simulation with the staggered grid finite difference in various kinds of complex geologic model.
This paper mainly studied the numerical simulation on horizontal earth surface, irregular surface layered media which have abruption, interlayer and other complex models, thought of the algorithm of feasibility and quality. The numerical experiment results indicate that this staggered grid finite difference could complete the elastic wave field simulation in various kinds of complex geologic model, weakened numerical dispersion markedly, had the characters of high simulation precision, strong commonality good stability and so on, could absorbing artificial boundary reflections effectively, attained good simulation effects of wave propagation. In addition, experiment had proved under the premise of promising precision, bigger mesh spacing could be used to improve calculative efficiency.
引文
[1]常君勇,孟小红.交错网格有限差分法地震数值模拟[D].北京：中国地质大学地球探测与信息技术专业,2007：1
[2]李桂花,朱光明.井间地震波场数值模拟及波场特征研究[D].西安：西安科技大学地质工程专业,2008：34
[3]李振春,刘庆敏.高阶差分数值模拟方法研究[D].北京：中国石油大学,2007
[4]Cerveny V., Molotkov I.A.Psencik. Ray Method in Seismology [M]. Charles Univ. Press, Prague, 1977
[5]朱光明,曹建章.高斯射线束合成记录[M].西安：西北工业大学出版社,1993
[6]Kosloff D., Baysal E.. Forward Modeling by a Fourier Method [J]. Geophysics,1982,47(10): 1954-1966.
[7]Dmith W. D., The Application of Finite-element Analysis to body wave propagation problems [J]. Geophysics. Roy. Astr. Soc.,1975,42:747-768.
[8]Kelly K R et al. Synthetic seismograms:A finite-difference approach [J]. Geophysics,1976,41: 2-27.
[9]安艺敬一,P.G.理查兹.定量地震学理论和方法[M].北京：地震出版社,1986
[10]Alterman Z. and Karak F.C.. Propagation of elastic wave in layered media by finite-difference methods [J]. Bull., Seism. Soc. Am.,1968,58:367-398.
[11]Boore D.M.. Finite-difference methods for seismic wave propagation in heterogeneous materials, in methods in computation physics [N].11:B. A. Bolt, ed., Academic Press, inc.
[12]Alford R.M., Kelly K.R. and Boore D.M.. Accuracy of finite-difference methods modeling of the acoustic wave equation [J]. Geophysics,1974,39:834-842.
[13]Mdariga R.. Dynamics of an expanding circular fault[J].1976, Bull., Seism. Soc. Am.,65,163-182.
[14]Virieux J. P-SH wave propagation in heterogeneous media:velocity-stress finite-difference method [J]. Geophysics,1984,49(11):1933-1957.
[15]Dablain M A. The application of high-differencing to the scalar wave equation [J]. Geophysics, 1986,51(1):54-66.
[16]Mufti I R. Large-scale three dimensional seismic models and their interpretive significance [J]. Geophysics,1990,55:1166-1182.
[17]Bayliss A, Jordan K E et al.. A fourth-order accurate finite-difference scheme for the computation of elastic waves [J]. Bull. Seism. Soc. Am.,1986,76:1115-1132.
[18]Levander A R. Four-order finite difference P-SV seismograms [J]. Geophysics.1988,53(11): 1425-1436.
[19]Crase E. High-order(space and tine)finite-difference modeling of the elastic equation.60th Ann. Internat. Mtg. Soc. Expl. Geophysics, Expanded, Abstracts,1990,987-991.
[20]董良国,马在田等.一阶弹性波方程交错网格高阶差分解法稳定性问题[J].地球物理学报,2000,43(6)：856-864.
[21]裴正林.任意起伏地表弹性波方程交错网格高阶有限差分法数值模拟[J].石油地球物理勘探,2004,39(6)：629-634.
[22]徐果明,林庆忠等.二维复杂介质的块状建模及射线追踪[J].石油地球物理勘探,2001,36(2)：213-219.
[23]普雷帕拉塔F P,沙莫斯M I,庄心谷译.计算几何导论[M].北京：科学出版社,1990
[21]朱怀球.Voronoi cells网格生成与C∞插值基函数的新算法和软件开发及其在计算流体力学中的应用研究[D].北京：北京大学力学与工程科学系,2000
[25]吴江航.湍流高精度计算方法研究[R].95攀登计划预选项目进展报告,1998
[26]朱怀球,吴江航.一种基于Voronoi cells的C∞插值基函数及其在计算流体力学中的若干应用.北京大学学报(自然科学版),第37卷,第5期,2001.9,669-678.
[27]Sambridge M, Braun J, McQueen H. Geophysical Parameterization and Interpolation of Irregular Data Using Natural Neighbors [J]. Geophysics. Int,1995,122:837-857.
[28]朱怀球,吴江航.基于C∞基函数的自然邻点插值(NNI)方法在科学计算可视化上的应用.北京：北京大学湍流研究国家重点实验室,计算机工程与应用,2001.1
[29]谭捍东,韩冰.利用25点有限差分方法对二维弹性波正演模拟[D].北京：中国地质大学,2007：1
[30]王祥春,刘学伟.起伏地表二维声波方程地震波场模拟与分析[J].石油地球物理勘探,2007,42(3)：268-276.
[31]张剑锋.弹性波数值模拟的非规则网格差分法[J].地球物理学报,第41卷,增刊,1998.6
[32]Zienkiewitz O C. The Finite Element Method. New York:McGraw Hill,1977
[33]Mufti L R. Seismic modeling in the implicit mode [J]. Geophysics. Prosp.,1985,33:619-656.
[34]Virieux J. P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method. Geophysics,1986,51(4):889-901.
[35]马在田.地震成像技术-有限差分法偏移[M].石油工业出版社.1989.8
[36]马在田,曹景忠,王家林等著.计算地球物理学概论[J].上海：同济大学出版社,1997,20-27.Ma Z T, Cao J Z, Wang J L, et al. Conspectus of Computational. Geophysics(in Chinese). Shanghai:Tongji University Press,1977,20-27.
[37]董国良,马在田等.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.
[38]周竹生,刘喜亮等.弹性介质中瑞雷面波有限差分法正演模拟[J].地球物理学报,2007,50(2)：567-573.
[39]王华忠,马在田.优化系数的吸收边界条件方程[J].石油地球物理勘探,1997,32(5)：628-636.
[40]罗大清,宋炜,吴律.一种有效的处理模型角点反射的方法[J].石油物探,2000,39(4)： 26-31.
[41]Smith, W. D..A non-reflecting plane boundary for wave propagation problems [J]. Comp. Phys..1974,15:492-503.
[42]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)：53-58.
[43]Clayton R and Engquist B. Absorbing boundary conditions for acoustic and elastic wave equations [J]. Bull Seis Soc Am,1977, (67):1529-1540.
[44]Engquist B and Majda A. Absorbing boundary conditions for the numerical simulation of waves [J]. Math Comp,1977, (31):629-651.
[45]Reynolds A C. Boundary conditions for the numerical solution of wave propagation problems [J]. Geophysics,1978,43:1099-1110.
[46]Keys RG. Absorbing boundary conditions for acoustic media[J]. Geophysics,1985, (50):892-902.
[47]Randall C J. Absorbing boundary condition for the elastic wave equation [J]. Geophysics,1988, (53):611-624.
[48]Long L T. and L iow J S.. A transparent boundary for finite-difference wave simulation. Geophysics, 1990, (27):831-870.
[49]梁非,杨慧珠.弹性介质波传问题的吸收边界条件[J].工程力学,1997.5,14(2)：120-127.
[50]Berenger J P. A perfectly matched layer for the absorption of electromagnetic waves [J]. Compote. Phys.,1994,114:185-200.
[51]张明,张晓丹.二维波动方程吸收边界问题的精细积分解法[J].高等学校计算数学学报.2005.12,第27卷专辑,132-138.
[52]董良国.弹性波数值模拟中的吸收边界条件[J].石油地球物理勘探.1999,34(1)：45-56.
[53]冯英杰,杨长春等.地震波有限差分模拟综述[J].地球物理学进展,2007,22(2)：487-491.
[54]Rodrigues D, Mora P. Analysis of a finite-difference solution to3-D elastic wave propagation.62nd J Ann. Internet, Mtg.Soc.Expl. Geophysics, Expanded Abstracts,1992,1247-1250.
[2]李桂花,朱光明.井间地震波场数值模拟及波场特征研究[D].西安：西安科技大学地质工程专业,2008：34
[3]李振春,刘庆敏.高阶差分数值模拟方法研究[D].北京：中国石油大学,2007
[4]Cerveny V., Molotkov I.A.Psencik. Ray Method in Seismology [M]. Charles Univ. Press, Prague, 1977
[5]朱光明,曹建章.高斯射线束合成记录[M].西安：西北工业大学出版社,1993
[6]Kosloff D., Baysal E.. Forward Modeling by a Fourier Method [J]. Geophysics,1982,47(10): 1954-1966.
[7]Dmith W. D., The Application of Finite-element Analysis to body wave propagation problems [J]. Geophysics. Roy. Astr. Soc.,1975,42:747-768.
[8]Kelly K R et al. Synthetic seismograms:A finite-difference approach [J]. Geophysics,1976,41: 2-27.
[9]安艺敬一,P.G.理查兹.定量地震学理论和方法[M].北京：地震出版社,1986
[10]Alterman Z. and Karak F.C.. Propagation of elastic wave in layered media by finite-difference methods [J]. Bull., Seism. Soc. Am.,1968,58:367-398.
[11]Boore D.M.. Finite-difference methods for seismic wave propagation in heterogeneous materials, in methods in computation physics [N].11:B. A. Bolt, ed., Academic Press, inc.
[12]Alford R.M., Kelly K.R. and Boore D.M.. Accuracy of finite-difference methods modeling of the acoustic wave equation [J]. Geophysics,1974,39:834-842.
[13]Mdariga R.. Dynamics of an expanding circular fault[J].1976, Bull., Seism. Soc. Am.,65,163-182.
[14]Virieux J. P-SH wave propagation in heterogeneous media:velocity-stress finite-difference method [J]. Geophysics,1984,49(11):1933-1957.
[15]Dablain M A. The application of high-differencing to the scalar wave equation [J]. Geophysics, 1986,51(1):54-66.
[16]Mufti I R. Large-scale three dimensional seismic models and their interpretive significance [J]. Geophysics,1990,55:1166-1182.
[17]Bayliss A, Jordan K E et al.. A fourth-order accurate finite-difference scheme for the computation of elastic waves [J]. Bull. Seism. Soc. Am.,1986,76:1115-1132.
[18]Levander A R. Four-order finite difference P-SV seismograms [J]. Geophysics.1988,53(11): 1425-1436.
[19]Crase E. High-order(space and tine)finite-difference modeling of the elastic equation.60th Ann. Internat. Mtg. Soc. Expl. Geophysics, Expanded, Abstracts,1990,987-991.
[20]董良国,马在田等.一阶弹性波方程交错网格高阶差分解法稳定性问题[J].地球物理学报,2000,43(6)：856-864.
[21]裴正林.任意起伏地表弹性波方程交错网格高阶有限差分法数值模拟[J].石油地球物理勘探,2004,39(6)：629-634.
[22]徐果明,林庆忠等.二维复杂介质的块状建模及射线追踪[J].石油地球物理勘探,2001,36(2)：213-219.
[23]普雷帕拉塔F P,沙莫斯M I,庄心谷译.计算几何导论[M].北京：科学出版社,1990
[21]朱怀球.Voronoi cells网格生成与C∞插值基函数的新算法和软件开发及其在计算流体力学中的应用研究[D].北京：北京大学力学与工程科学系,2000
[25]吴江航.湍流高精度计算方法研究[R].95攀登计划预选项目进展报告,1998
[26]朱怀球,吴江航.一种基于Voronoi cells的C∞插值基函数及其在计算流体力学中的若干应用.北京大学学报(自然科学版),第37卷,第5期,2001.9,669-678.
[27]Sambridge M, Braun J, McQueen H. Geophysical Parameterization and Interpolation of Irregular Data Using Natural Neighbors [J]. Geophysics. Int,1995,122:837-857.
[28]朱怀球,吴江航.基于C∞基函数的自然邻点插值(NNI)方法在科学计算可视化上的应用.北京：北京大学湍流研究国家重点实验室,计算机工程与应用,2001.1
[29]谭捍东,韩冰.利用25点有限差分方法对二维弹性波正演模拟[D].北京：中国地质大学,2007：1
[30]王祥春,刘学伟.起伏地表二维声波方程地震波场模拟与分析[J].石油地球物理勘探,2007,42(3)：268-276.
[31]张剑锋.弹性波数值模拟的非规则网格差分法[J].地球物理学报,第41卷,增刊,1998.6
[32]Zienkiewitz O C. The Finite Element Method. New York:McGraw Hill,1977
[33]Mufti L R. Seismic modeling in the implicit mode [J]. Geophysics. Prosp.,1985,33:619-656.
[34]Virieux J. P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method. Geophysics,1986,51(4):889-901.
[35]马在田.地震成像技术-有限差分法偏移[M].石油工业出版社.1989.8
[36]马在田,曹景忠,王家林等著.计算地球物理学概论[J].上海：同济大学出版社,1997,20-27.Ma Z T, Cao J Z, Wang J L, et al. Conspectus of Computational. Geophysics(in Chinese). Shanghai:Tongji University Press,1977,20-27.
[37]董国良,马在田等.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.
[38]周竹生,刘喜亮等.弹性介质中瑞雷面波有限差分法正演模拟[J].地球物理学报,2007,50(2)：567-573.
[39]王华忠,马在田.优化系数的吸收边界条件方程[J].石油地球物理勘探,1997,32(5)：628-636.
[40]罗大清,宋炜,吴律.一种有效的处理模型角点反射的方法[J].石油物探,2000,39(4)： 26-31.
[41]Smith, W. D..A non-reflecting plane boundary for wave propagation problems [J]. Comp. Phys..1974,15:492-503.
[42]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)：53-58.
[43]Clayton R and Engquist B. Absorbing boundary conditions for acoustic and elastic wave equations [J]. Bull Seis Soc Am,1977, (67):1529-1540.
[44]Engquist B and Majda A. Absorbing boundary conditions for the numerical simulation of waves [J]. Math Comp,1977, (31):629-651.
[45]Reynolds A C. Boundary conditions for the numerical solution of wave propagation problems [J]. Geophysics,1978,43:1099-1110.
[46]Keys RG. Absorbing boundary conditions for acoustic media[J]. Geophysics,1985, (50):892-902.
[47]Randall C J. Absorbing boundary condition for the elastic wave equation [J]. Geophysics,1988, (53):611-624.
[48]Long L T. and L iow J S.. A transparent boundary for finite-difference wave simulation. Geophysics, 1990, (27):831-870.
[49]梁非,杨慧珠.弹性介质波传问题的吸收边界条件[J].工程力学,1997.5,14(2)：120-127.
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