非线性P波在二维固体中的传播特点数值模拟
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摘要
波动理论一直是地震学的核心内容,非线性波在介质中是真实存在的,能够充分认识非线性波,将对波动理论的发展和进一步应用提供很大的帮助.本文从固体介质中的位移、应力、应变的基本关系出发,推导了非线性P波在二维固体中的传播的非线性方程,对此非线性方程,采用四阶差分的方法,研究其在二维空间中的传播特点,对非线性波的传播有了更深的认识和理解.
Wave theory is the kernel content of seismology,and the nonlinear wave indeed exists in the solid,which will be helpful for the development of the wave theory and the further application if we can completely recognise it. This paper deductes the nonlinear P-wave equation in the 2-D solid based on the basic relation of the displacement, stress and strain.To this equation,using the 4-order difference method,we study the propagation characters and obtain further cognition and comprehension of the nonlinear P-wave in solid.
Wave theory is the kernel content of seismology,and the nonlinear wave indeed exists in the solid,which will be helpful for the development of the wave theory and the further application if we can completely recognise it. This paper deductes the nonlinear P-wave equation in the 2-D solid based on the basic relation of the displacement, stress and strain.To this equation,using the 4-order difference method,we study the propagation characters and obtain further cognition and comprehension of the nonlinear P-wave in solid.
引文
[1] 刘财,冯晅,何敏,等.非线性波动理论在地震勘探中的应用 综述[J].地球物理学进展,2000,15(4) ;68～78． Liu C,Feng X,He M,et al.Applying nonlinear wave theory to seismic exploration[J].Progress in Geophysics.(in Chinese),2000,15(4) :68～78．
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[3] Gist G A.Fluid effects on velocity and attenuation in sandstones[J].Acous Soc Amer,1994,96:1158～1173．
[4] Johnson P A,Shankland T J,O'Connell R J,et al.Nonlinear generation of elastic waves in crystalline rocks[J].Geophys Res,1987,92:3597～3602．
[5] Johnson P A,Shankland T J.Nonlinear generation of elastic waves in granite and sandstone:Continuous wave and travel time observations[J].Geophys Res,1989,94:17729～17733．
[6] 冯晅,杨宝俊,郑海山.非线性波动的几个基本问题[J].吉林大学学报:地球科学版,2002,32(2) :187～192． Feng X,Yang B J,Zheng H S.Some basic problems and analysis about nonlinear wave[J].Journal of Jilin university(Earth science edition).(in Chinese),2002,32(2) :187～192．
[7] 牛滨华,孙春岩.地震波理论研究进展--介质模型和地震波传播[J].地球物理学进展,2004,19(2) :255～263． Niu B H,Sun C Y.Developing theory of propagation of seismic waves--medium model and propagation of seismic waves[J].Progress in Geophysics.(in Chinese),2004,19(2) :255～263．
[8] 周竹生,刘喜亮,熊孝雨.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． Zhou Z S,Liu X L,Xiong X Y.Finite-difference modelling of Rayleigh surface wave in elastic media[J].Chinese J.Geophys.(in Chinese),2007,50(2) :567～573
[9] 张金海,王卫民,赵连锋,等.傅里叶有限差分法三维波动方程正演模拟[J].地球物理学报,2007,50(6) :1854～1862． Zhang J H,Wang W M,Zhao L F,et al.Modeling 3-D scalar waves using the Fourier finite difference method[J].Chinese J.Geophys.(in Chinese).,2007,50(6) :1854～1862．
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[11] 程冰洁,李小凡.2． 5维地震波场褶积微分算子法数值模拟[J].地球物理学进展,2008,23(4) :1099～1105． Cheng B J,Li X F.2． 5D modelling of elastic waves using the convolutional differentiator method[J].Progress in geophysics.(in Chinese),2008,23(4) :1099～1105．
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[13] 吴国忱,梁锴.VTI介质qP波方程高精度有限差分算子[J].地球物理学进展,2007,22(3) :896～904． Wu G C,Liang K.High precision finite difference operators for qP wave equation in VTI media[J].Progress in geophysics.(in Chinese),2007,22(3) :896～904．
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[15] 邵志刚,傅容珊,黄建华,等.线性粘弹介质中地震波场数值模拟[J].地球物理学进展,2007,22(4) :1135～1141． Shao Z G,Fu R S,H uang J H,et al.Modeling seismic wave in linear viscoelastic media[J].Progress in geophysics.(in Chinese),2007,22(4) :1135～1141．
[16] 郭桂红,石双虎,剡慧君,等.基于二维三分量伪谱法模拟数据的EDA介质中横波分裂研究[J].地球物理学6,92008,51(2) :469～478． Guo G H,Shi S H,Yan H J,et al.Azimuth-dependence of shear-wave splitting in EDA media:Two-dimensional threecomponent pseudospeetral modeling[J].Chinese J.Geophys.(in Chinese),2008,51(2) :469～478．
[17] 张素芳,张智.均匀介质背景中三维异常体的面波波场响应及其动力学特征分析[J].地球物理学报,2008,51(4) :1180～1187． Zhang S F,Zhang Z.Surface wave wavefield and dynamic analysis for three-dimensional abnormal structures with homegenous background velocity model[J].Chinese J.Geophys.(in Chinese),2008,51(4) :1180～1187．
[18] 孙银行.弱各向异性介质弹性波的准各向同性近似正演模拟[J].地球物理学进展,2008,23(.4) :1118～1123． Sun Y H.Elastic wave forward modelong with quasi-isotropic approximation in weakly anisotropic medium[J].Progress in Geophysics.(in Chinese),2008,23(4) :1118～1123．
[19] 钱祖文.非线性声学[M].北京:科学出版社,1992:324～389． Qian Z W.Nonlinear acoustics[M].Beijing:Science Press.(in Chinese),1992:324～389．
[20] 范尚武.地球反射资料构造成像的高阶差分方法[J].石油地球物理勘探,1995,30(4) :454～464． Fan S W.High-order difference method for structure image of seismic reflection data[J].Oil Geophysical Prospecting.(in Chinese),1995,30(4) :454～464．
[21] 高殿武,席时桐.高阶差分格式的离散方法及其应用研究[J].黑龙江矿业学院学报,1997,7(1) :68～72． Gao D W,Xi S T.A study on straggling method and application of high-order difference schemes[J].Journal of Heilongjiang institute of science.(in Chinese),1997,7(1) :68～72．
[22] 黄东涛,边晓东,刘克,等.非线性驻波现象的数值模拟与实验结果的比较[J].声学学报,1999,24(3) :295～303． Huang D T,Bian X D,Liu K,et al.A comparison of numerical simulation with experimental results for nonlinear standing wave phenomena[J].Acta Acustica.(in Chinese),1999,24(3) :295～303．
[23] 郑海山,张中杰,杨宝俊.固体中非线性纵波一维模拟[刀.地震学报,2004,26(1) :77～83． Zheng H S,Zhao Z J,Yang B J.A numerical study of 1-D nonlinear P-wave propagation in solid[J].Acta Seismologica Sinica.(in Chinese),2004,26(1) :77～83．
[24] 王兆国,秦树洪,颜军.固体介质中非线性波的能量初探[J].地球物理学进展,2007,22(1) :192～199． Wang Z G,Qin S H,Yan J.A study on energy of nonlinear wave in solid media[J].Progress in Geophysics.(in Chinese),2007,22(1) :192～199
[2] Toksoz M N,Cheng C H,Timur A.Velocities of seismic waves in porous rocks[J].Geophysics,1976,41:621～645．
[3] Gist G A.Fluid effects on velocity and attenuation in sandstones[J].Acous Soc Amer,1994,96:1158～1173．
[4] Johnson P A,Shankland T J,O'Connell R J,et al.Nonlinear generation of elastic waves in crystalline rocks[J].Geophys Res,1987,92:3597～3602．
[5] Johnson P A,Shankland T J.Nonlinear generation of elastic waves in granite and sandstone:Continuous wave and travel time observations[J].Geophys Res,1989,94:17729～17733．
[6] 冯晅,杨宝俊,郑海山.非线性波动的几个基本问题[J].吉林大学学报:地球科学版,2002,32(2) :187～192． Feng X,Yang B J,Zheng H S.Some basic problems and analysis about nonlinear wave[J].Journal of Jilin university(Earth science edition).(in Chinese),2002,32(2) :187～192．
[7] 牛滨华,孙春岩.地震波理论研究进展--介质模型和地震波传播[J].地球物理学进展,2004,19(2) :255～263． Niu B H,Sun C Y.Developing theory of propagation of seismic waves--medium model and propagation of seismic waves[J].Progress in Geophysics.(in Chinese),2004,19(2) :255～263．
[8] 周竹生,刘喜亮,熊孝雨.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． Zhou Z S,Liu X L,Xiong X Y.Finite-difference modelling of Rayleigh surface wave in elastic media[J].Chinese J.Geophys.(in Chinese),2007,50(2) :567～573
[9] 张金海,王卫民,赵连锋,等.傅里叶有限差分法三维波动方程正演模拟[J].地球物理学报,2007,50(6) :1854～1862． Zhang J H,Wang W M,Zhao L F,et al.Modeling 3-D scalar waves using the Fourier finite difference method[J].Chinese J.Geophys.(in Chinese).,2007,50(6) :1854～1862．
[10] 郭智奇,刘财,杨宝俊,等.粘弹各向异性介质中地震波场模拟与特征[J].地球物理学进展,2007,22(3) :804～810． Guo Z Q,Liu C,Yang B J,et al.Seismic wave fields modeling and feature in viscoelastic anisotropic media[J].Progress in geophysics.(in Chinese),2007,22(3) :804～810．
[11] 程冰洁,李小凡.2． 5维地震波场褶积微分算子法数值模拟[J].地球物理学进展,2008,23(4) :1099～1105． Cheng B J,Li X F.2． 5D modelling of elastic waves using the convolutional differentiator method[J].Progress in geophysics.(in Chinese),2008,23(4) :1099～1105．
[12] 武晔,李小凡,顾观文,等.2． 5维非均匀介质中的地震波数值模拟[J].地球物理学进展,2008,23(4) ,1092～1098． Wu Y,Li X F,Gu G W,et al.2． 5D numerical simulation of seismic wave in inhomogeneous media[J].Progress in geophysics.(in Chinese),2008,23(4) ,1092～1098．
[13] 吴国忱,梁锴.VTI介质qP波方程高精度有限差分算子[J].地球物理学进展,2007,22(3) :896～904． Wu G C,Liang K.High precision finite difference operators for qP wave equation in VTI media[J].Progress in geophysics.(in Chinese),2007,22(3) :896～904．
[14] 窦玉坛,郭常升,毛中华,等.井间地震波波动方程正演数值模拟及其应用[J].地球物理学进展,2007,22(6) :1867～1872． Dou Y T,Guo C S,Mao Z H,et al.The wave equation forward numerical simulation and application in crosswell seismic[J].Progress in geophysics.(in Chinese),2007,22(6) :1867～1872．
[15] 邵志刚,傅容珊,黄建华,等.线性粘弹介质中地震波场数值模拟[J].地球物理学进展,2007,22(4) :1135～1141． Shao Z G,Fu R S,H uang J H,et al.Modeling seismic wave in linear viscoelastic media[J].Progress in geophysics.(in Chinese),2007,22(4) :1135～1141．
[16] 郭桂红,石双虎,剡慧君,等.基于二维三分量伪谱法模拟数据的EDA介质中横波分裂研究[J].地球物理学6,92008,51(2) :469～478． Guo G H,Shi S H,Yan H J,et al.Azimuth-dependence of shear-wave splitting in EDA media:Two-dimensional threecomponent pseudospeetral modeling[J].Chinese J.Geophys.(in Chinese),2008,51(2) :469～478．
[17] 张素芳,张智.均匀介质背景中三维异常体的面波波场响应及其动力学特征分析[J].地球物理学报,2008,51(4) :1180～1187． Zhang S F,Zhang Z.Surface wave wavefield and dynamic analysis for three-dimensional abnormal structures with homegenous background velocity model[J].Chinese J.Geophys.(in Chinese),2008,51(4) :1180～1187．
[18] 孙银行.弱各向异性介质弹性波的准各向同性近似正演模拟[J].地球物理学进展,2008,23(.4) :1118～1123． Sun Y H.Elastic wave forward modelong with quasi-isotropic approximation in weakly anisotropic medium[J].Progress in Geophysics.(in Chinese),2008,23(4) :1118～1123．
[19] 钱祖文.非线性声学[M].北京:科学出版社,1992:324～389． Qian Z W.Nonlinear acoustics[M].Beijing:Science Press.(in Chinese),1992:324～389．
[20] 范尚武.地球反射资料构造成像的高阶差分方法[J].石油地球物理勘探,1995,30(4) :454～464． Fan S W.High-order difference method for structure image of seismic reflection data[J].Oil Geophysical Prospecting.(in Chinese),1995,30(4) :454～464．
[21] 高殿武,席时桐.高阶差分格式的离散方法及其应用研究[J].黑龙江矿业学院学报,1997,7(1) :68～72． Gao D W,Xi S T.A study on straggling method and application of high-order difference schemes[J].Journal of Heilongjiang institute of science.(in Chinese),1997,7(1) :68～72．
[22] 黄东涛,边晓东,刘克,等.非线性驻波现象的数值模拟与实验结果的比较[J].声学学报,1999,24(3) :295～303． Huang D T,Bian X D,Liu K,et al.A comparison of numerical simulation with experimental results for nonlinear standing wave phenomena[J].Acta Acustica.(in Chinese),1999,24(3) :295～303．
[23] 郑海山,张中杰,杨宝俊.固体中非线性纵波一维模拟[刀.地震学报,2004,26(1) :77～83． Zheng H S,Zhao Z J,Yang B J.A numerical study of 1-D nonlinear P-wave propagation in solid[J].Acta Seismologica Sinica.(in Chinese),2004,26(1) :77～83．
[24] 王兆国,秦树洪,颜军.固体介质中非线性波的能量初探[J].地球物理学进展,2007,22(1) :192～199． Wang Z G,Qin S H,Yan J.A study on energy of nonlinear wave in solid media[J].Progress in Geophysics.(in Chinese),2007,22(1) :192～199
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