二维SH波方程的半解析解及其数值模拟
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摘要
本文以波动理论为基础,半解析化求解地震勘探中常用的SH波方程.获得的主要结果包括:给出了二维均匀介质中SH波方程的解析解;利用Cagniard-de Hoop方法详细推导了二维双层介质中SH波方程的解析解,获得了透射波的解析解表达式.同时,基于SH波方程的解析表达式,给出了包含各种波(如直达波、反射波、首波以及透射波)的解析解和波形图.对于比较复杂的积分型解析解,利用数值积分方法给出了数值结果,并与优化的近似解析离散化方法(ONADM)和4阶Lax-Wendroff修正方法(LWC)的数值结果进行了比较,以验证解析解的正确性.本文的研究成果有望在检验波动方程数值新方法的有效性、波传播理论分析等方面得到应用.
In this paper,we show the analytical solutions of the SH-wave equation based on wave propagation theories.The main results include the analytical solutions of 2D SH-wave equations in the homogeneous medium,the analytical solutions of reflected SH-wave in a two-layer medium derived in detail by using the Cagniard-de Hoop method;we also obtain the analytical solutions for the transmitted SH-wave.Meanwhile,we present some waveforms of various waves including direct wave,reflected wave,head wave,and transmitted wave computed by the analytical solutions of the SH-wave.We present some numerical results by using numerical integral algorithms for the complex integral solutions,and compare the analytical solutions with the numerical results computed by the numerical methods including the optimal nearly analytical discrete method(ONADM) and the fourth-order Lax-Wendroff correction(LWC) scheme to verify the correctness of the numerical methods.The analytical solutions obtained in this paper have great potentials in the applications of testing the new methods for solving the wave equations and the theoretical analysis of wave propagation.
In this paper,we show the analytical solutions of the SH-wave equation based on wave propagation theories.The main results include the analytical solutions of 2D SH-wave equations in the homogeneous medium,the analytical solutions of reflected SH-wave in a two-layer medium derived in detail by using the Cagniard-de Hoop method;we also obtain the analytical solutions for the transmitted SH-wave.Meanwhile,we present some waveforms of various waves including direct wave,reflected wave,head wave,and transmitted wave computed by the analytical solutions of the SH-wave.We present some numerical results by using numerical integral algorithms for the complex integral solutions,and compare the analytical solutions with the numerical results computed by the numerical methods including the optimal nearly analytical discrete method(ONADM) and the fourth-order Lax-Wendroff correction(LWC) scheme to verify the correctness of the numerical methods.The analytical solutions obtained in this paper have great potentials in the applications of testing the new methods for solving the wave equations and the theoretical analysis of wave propagation.
引文
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[7]Levander A R.Fourth-order finite-difference P-SV seismograms.Geophysics,1988,53(11):1425-1436.
[8]Komatitsch D,Barnes C,Tromp J.Simulation of anisotropicwave propagation based upon a spectral element method.Geophysics,2000,65(4):1251-1260.
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[12]Bhattacharya S N.Exact solutions of SH wave equation intransversely isotropic inhomogeneous elastic media.Pure andAppl.Geophys.,1972,93(1):19-35.
[13]Aki K,Richards P G.Quantitative Seismology.Sausalito,Calif.:University Science Books,2002.
[14]杨桂通,张善元.弹性动力学.北京:中国铁道出版社,1988.Yang G T,Zhang S Y.Elastodynamics(in Chinese).Beijing:China Railway Press,1988
[15]Zhou H,Chen X F.Ray path of head waves with irregularinterfaces.Applied Geophysics,2010,7(1):66-73.
[16]Cerveny V,Ravindra R.Theory of Seismic Head Waves.Toronto:University of Toronto Press,1971.
[17]李庆扬,王能超,易大义.数值分析.北京:清华大学出版社,2001.Li Q Y,Wang N C,Yi D Y.Numerical Analysis(inChinese).Beijing:Tsinghua University Press,2001.
[2]Yang D H,Teng J W,Zhang Z J,et al.A nearly analyticdiscrete method for acoustic and elastic wave equations inanisotropic media.Bull.Seism.Soc.Am.,2003,93(2):882-890.
[3]Yang D H,Peng J M,Lu M,et al.Optimal nearly analyticdiscrete approximation to the scalar wave equation.Bull.Seism.Soc.Am.,2006,96(3):1114-1130.
[4]杨顶辉.双相各向异性介质中弹性波方程的有限元解法及波场模拟.地球物理学报,2002,45(4):575-583.Yang D H.Finite element method of the elastic waveequation and wavefield simulation in two-phase anisotropicmedia.Chinese J.Geophys.(in Chinese),2002,45(4):575-583.
[5]王妙月,郭亚曦,底青云.二维线性流变体波的有限元模拟.地球物理学报,1995,38(4):494-506.Wang M Y,Guo Y X,Di Q Y.2-D finite element modellingfor seismic wave in media with linear rheological property.Chinese J.Geophys.(in Chinese),1995,38(4):494-506.
[6]邓玉琼,张之立.弹性波的三维有限元模拟.地球物理学报,1990,33(1):44-53.Deng Y Q,Zhang Z L.Three-dimensional finite elementmodelling of elastic waves.Chinese J.Geophys.(in Chinese),1990,33(1):44-53.
[7]Levander A R.Fourth-order finite-difference P-SV seismograms.Geophysics,1988,53(11):1425-1436.
[8]Komatitsch D,Barnes C,Tromp J.Simulation of anisotropicwave propagation based upon a spectral element method.Geophysics,2000,65(4):1251-1260.
[9]Ewing W M,Jardetzky W S,Press F.Elastic Waves inLayered Media.New York:McGraw-Hill Book CompanyInc.,1957.
[10]de Hoop A T.A modification of Cagniard's method forsolving seismic pulse problems.Appl.sci.Res.(section B),1960,8(1):349-356.
[11]Bhattacharya S N.Exact solutions of SH wave equation forinhomogeneous media.Bull.Seism.Soc.Am.,1970,60(6):1847-1859.
[12]Bhattacharya S N.Exact solutions of SH wave equation intransversely isotropic inhomogeneous elastic media.Pure andAppl.Geophys.,1972,93(1):19-35.
[13]Aki K,Richards P G.Quantitative Seismology.Sausalito,Calif.:University Science Books,2002.
[14]杨桂通,张善元.弹性动力学.北京:中国铁道出版社,1988.Yang G T,Zhang S Y.Elastodynamics(in Chinese).Beijing:China Railway Press,1988
[15]Zhou H,Chen X F.Ray path of head waves with irregularinterfaces.Applied Geophysics,2010,7(1):66-73.
[16]Cerveny V,Ravindra R.Theory of Seismic Head Waves.Toronto:University of Toronto Press,1971.
[17]李庆扬,王能超,易大义.数值分析.北京:清华大学出版社,2001.Li Q Y,Wang N C,Yi D Y.Numerical Analysis(inChinese).Beijing:Tsinghua University Press,2001.
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