基于广义正交多项式褶积微分算子的地震波场数值模拟方法
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摘要
地震波场模拟方法研究对于与波动现象有关的地震学问题的重要性是不言而喻的.就目前现有的各种正演算法来说,精度较高的算法(如有限元法、谱元法、高阶有限差分法等),其计算速度较慢;计算速度较快的算法(如低阶有限差分法、付氏伪谱法等)计算精度却比较低.为了兼顾地震波场模拟的精度与速度,本文推出了一种快速的、高精度地震波场模拟方法(基于Forsyte广义正交多项式的褶积微分算子法),该方法是以计算数学中的Forsyte广义正交多项式插值函数为基础,构建一个新的褶积微分算子,并将该算子引入到地震波动方程的一阶速度-应力方程的空间微分运算中去,采用时间交错网格有限差分算子替代普通的差分算子以匹配高精度的褶积微分算子,从而构造一种全新的地震波场数值模拟方法.该方法同时具有广义正交多项式方法的高精度和短算子低阶有限差分算法的高速度.通过对算子长度的调节及算子系数的优化,可同时兼顾波场解的全局信息与局部信息.复杂非均匀介质模型中的波场数值模拟实验证实了该方法的可行性及优越性.
Numerical modeling is important in studying a variety of problems in seismic wave propagation, many methods have been investigated for calculating the spatial terms in the wave equations, the most widely used methods are the classical finite difference (FD) method, the Fourier pseudo-spectral (FPS) method and Finite Element (FE) method. It is clear that each method has its own merits and drawbacks. For improving the precision and efficiency of seismic modeling, this paper develops a new modeling approach (Convolutional Forsyte Polynomial Differentiator Method) by using optimized convolutional operators for spatial differentiation and staggered-grid finite-difference for time differentiation in wave equation computation, a theoretical computation example of 2-D seismic wave field is given and the numerical results show that the algorithm can bring reliable outcomes with high precision and fast speed, and be adapted to complex inhomogeneous geological models and readily be extended to wave modeling for anisotropic media.
Numerical modeling is important in studying a variety of problems in seismic wave propagation, many methods have been investigated for calculating the spatial terms in the wave equations, the most widely used methods are the classical finite difference (FD) method, the Fourier pseudo-spectral (FPS) method and Finite Element (FE) method. It is clear that each method has its own merits and drawbacks. For improving the precision and efficiency of seismic modeling, this paper develops a new modeling approach (Convolutional Forsyte Polynomial Differentiator Method) by using optimized convolutional operators for spatial differentiation and staggered-grid finite-difference for time differentiation in wave equation computation, a theoretical computation example of 2-D seismic wave field is given and the numerical results show that the algorithm can bring reliable outcomes with high precision and fast speed, and be adapted to complex inhomogeneous geological models and readily be extended to wave modeling for anisotropic media.
引文
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[18]Kosloff R.Absorbing boundaries for wave propagation problems.J.Comput.Phys.,1986,63:363~376
[19]Zhou B,Greenhalgh S A.Seismic scalar wave equation modeling by a convolutional differentiator.Bulletin ofthe Seismological Society of America,1992,82(1):289~303
[20]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1):63~69Zhang Z J,Teng J W,Yang D H.Acoustic and elastic wave modeling by a convolutional differentiator.Acta Seismologica Sinica(in Chinese),1996,18(1):63~69
[2]裴正林,牟永光.地震波传播数值模拟.地球物理学进展,2004,19(4):933~941Pei Z L,Mou Y G.Numerical simulation of seismic wave propagation.Progress in Geophysics(in Chinese):2004,19(4):933~941
[3]郑洪伟,李廷栋,高锐等.数值模拟在地球动力学中的研究进展.地球物理学进展,2006,21(2):360~369Zheng H W,Li T D,Gao R,et al.The advance of numerical simulationin geodynamics.Progress in Geophysics(in Chinese),2006,21(2):360~369
[4]Chen Kunhua.Simulation of seismic propagation in anisotropic media:finite element method.The54th SEG Annual Meeting Expanded Abstract,Houston,USA,1984,631~632
[5]周辉,徐世浙,刘斌等.各向异性介质中波动方程有限元法模拟及其稳定性.地球物理学报,1997,40(6):833~841Zhou H,Xu S Z,Liu B,et al.Modeling of wave equations in anisotropic media using finite element method and its stability condition.Chinese J.Geophys.(in Chinese),1997,40(6):833~841
[6]张美根,王妙月,李小凡.各向异性弹性波场的有限元数值模拟.地球物理学报,2002,17(3):384~413Zhang MG,Wang MY,Li XF.Finite element forward modeling of anisotropic elastic waves.Chinese J.Geophys.(in Chinese),2002,17(3):384~413
[7]Komatitsch D,Tromp J.Introductionto the spectral-element method for3-Dseismic wave propagation.Geophys.J.Int.,1999,139:806~822
[8]马德堂,朱光明.有限元法与伪谱法混和求解弹性波动方程.地球科学与环境学报,2004,26(1):61~64Ma D T,Zhu G M.Hybrid method combining finite element and pseudo-spectral methodfor solvingthe elastic wave equation.Journal of Earth Sciences and Environment(in Chinese),2004,26(1):61~64
[9]Crase E.High-order(space and time)finite difference modeling of elastic wave equation.Expanded Abstacts of60th SEG Annual Meeting,1990.987~991
[10]董国良,马在田,曹景忠等.一阶弹性波方程交错网格高阶差分解法.地球物理学报,2000,43(3):411~419Dong G L,Ma Z T,Cao J Z,et al.Astaggered-grid high-order difference method of one-order elastic wave equation.Chinese J.Geophys.(in Chinese),2000,43(3):411~419
[11]Fornberg B.The pseudo-spectral method:comparisons with finite differences for the elastic wave equation.Geophysics,1987,52(4):483~501
[12]金胜汶,陈必远,马在田.三维波动方程有限差分正演方法.地球物理学报,1994,37(6):804~810Jin S W,Chen B Y,Ma Z T.Three dimensional wave equation forward modeling by the finite difference method.Chinese J.Geophys.(in Chinese),1994,37(6):804~810
[13]Kosloff D,Baysal E.Forward modeling by a Fourier method.Geophysics,1982,47(10):1402~1412
[14]赵志新,徐纪人,堀内茂木等.交错网格实数傅里叶变换微分算子及其在非均匀介质波动传播研究中的应用.地球物理学报,2003,46(2):234~240Zhao Z X,Xu J R,Shigeki Horiucki,et al.Staggered-grid real value FFT differentiation operator and its application on wave propagation simulation in the heterogeneous medium.Chinese J.Geophys.(in Chinese),2003,46(2):234~240
[15]谢靖.物探数据处理的数学方法(下册).北京:地质出版社,1981.97~104Xie J.Numerical Methodfor Geophysical Data Process(in Chinese).Beijing:Geological Publishing House,1981.97~104
[16]Gazdag J.Modelling of the acoustic wave equation with transform method.Geophysics,1981,46:854
[17]Cerjan C,Kosloff D,Kosolff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation.Geophysics,1985,50:705~708
[18]Kosloff R.Absorbing boundaries for wave propagation problems.J.Comput.Phys.,1986,63:363~376
[19]Zhou B,Greenhalgh S A.Seismic scalar wave equation modeling by a convolutional differentiator.Bulletin ofthe Seismological Society of America,1992,82(1):289~303
[20]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法.地震学报,1996,18(1):63~69Zhang Z J,Teng J W,Yang D H.Acoustic and elastic wave modeling by a convolutional differentiator.Acta Seismologica Sinica(in Chinese),1996,18(1):63~69
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