三维TTI介质相速度和群速度
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摘要
相速度和群速度是研究地震波传播规律和描述介质特性的重要参数,是弹性波传播理论中的核心内容,在理论研究和实际应用中有重要作用.本文根据VTI介质的刚度矩阵,利用Bond变换建立了TTI介质刚度矩阵.再利用TTI介质刚度矩阵,结合弹性动力学的本构方程、牛顿运动微分方程和几何方程,得到了三维TTI介质弹性波波动方程和Christoffel方程.通过本征值方法求解Christoffel方程,推导了三维TTI介质弹性波相速度的解析表达式.利用Berryman和Crampin推导各向异性介质群速度公式,根据三维TTI介质的相速度解析式推导了三维TTI介质群速度解析表达式.数值试例表明,随着各向异性介质参数改变,TI介质弹性波相速度变化较为平缓,群速度变化较为剧烈,qP波和SH波速度变化较为平缓,qSV波速度变化较为剧烈.
Phase and group velocity are the important parameters for researching seismic wave propagation and describing the media property,and they are key contents in theory of elastic wave propagation and play a role in theory and application.Based on stiffness matrix in vertical transverse isotropy (VTI)media and Bond transform, this paper found the stiffness matrix in tilted transverse isotropy (TTI)media.Using the stiffness matrix in TTI media,constitutive equation,Newton's motion differential equation and geometry equation,we obtain the elastic wave equation and Christoffel equation in 3D TTI media.Solving the Christoffel equation by eigenvalue method,we derive an analytic expression for phase velocity in 3D TTI media.On the basis of the group velocity equation derived by Berryman and Crampin,we derive an analytic expression for group velocity in 3D TTI media from the analytic expression for phase velocity.The result of numerical examples indicates that the change of elastic wave phase velocity in TI media is smoothout with anisotropic parameters,and the change of group velocity is acuteness.The change of velocity for qP wave and SH wave is smoothout,and the change of velocity for qSV wave is acuteness.
Phase and group velocity are the important parameters for researching seismic wave propagation and describing the media property,and they are key contents in theory of elastic wave propagation and play a role in theory and application.Based on stiffness matrix in vertical transverse isotropy (VTI)media and Bond transform, this paper found the stiffness matrix in tilted transverse isotropy (TTI)media.Using the stiffness matrix in TTI media,constitutive equation,Newton's motion differential equation and geometry equation,we obtain the elastic wave equation and Christoffel equation in 3D TTI media.Solving the Christoffel equation by eigenvalue method,we derive an analytic expression for phase velocity in 3D TTI media.On the basis of the group velocity equation derived by Berryman and Crampin,we derive an analytic expression for group velocity in 3D TTI media from the analytic expression for phase velocity.The result of numerical examples indicates that the change of elastic wave phase velocity in TI media is smoothout with anisotropic parameters,and the change of group velocity is acuteness.The change of velocity for qP wave and SH wave is smoothout,and the change of velocity for qSV wave is acuteness.
引文
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[18] 牛滨华,何樵登.孙春岩.六方各向异性介质方位矢量波动方程及其相速度[J].石油物探,1 994,33(1) :19～29． Niu B H,He Q D,Sun C Y.Azimuth vector wave equation and its phase velocity in hexagonal anisotropy medium[J].GPP,1994,33(1) :19～29．
[19] 郝重涛,姚陈,王迅.任意空间取向TI介质中速度随方位变化特征[J].地球物理学进展,2006,21(2) :524～530． Hao C T,Yao C,Wang X.The characteristics of velocities with azimuth variation for arbitrary spatial orientation TI media[J].Progress in Geophysics,2006,21(2) :524～530．
[20] 郝重涛,姚陈.任意空间取向TI介质中体波速度特征[J]. 地球物理学报,2007,50(2) :546～555． Hao C T,Yao C.Analysis of bodywave velocity characteristic for TI medium with arbitrary spatial orientation [J].ChineseJ.Geophys.(in Chinese),2007,50(2) :546～555．
[21] 赵爱华,丁志峰.一种弱各向异性介质地震波群速度的近似表示新方法[J].地球物理学进展,2005,20(4) :916～919． Zhao A H,Ding Z F.Ne w approximate expressions of Seismic group velocities for weakly anisotropic media [J].Progress in Geophysics,2005,20(4) :916～919．
[22] Xuan Y H,He Q D,Lin Y.Phase velocity in 2D TTI media [J].Applied Geophysics,2007,4(1) :25～28．
[23] Winterstein D F.Velocity anisotropy terminology for geophysicists[J].Geophysics,55(8) :1070～1088．
[24] Berryman J G.Long-wave elastic anisotropy in transversely isotropic media[J].Geophysics,1979,44(5) :896～917．
[2] Thomsen L.Weak elastic anisotropy[J].Geophysics,1986,51(10) :1954～1966．
[3] Alkbalifah T.An acoustic wave equation for anisotropic media [J].Geophysics,2000,65(4) :1239～1250．
[4] Alkhalifah T.Acoustic approximation for processing in transversely isotropic media[J].Geophysics,1998,63 (2) :623～631．
[5] Tsvankin I.Moveout analysis for transversely isotropic media with a tilted symmetry axis [J].Geophysical Prospecting,1997,45(3) ,479～512．
[6] Zhang LB,RectorJW Ⅲ,HoverstenG M.An acoustic wave equation for modeling in tilted TI media.73rd Ann.Internat.Mtg.,Soc.Expl.Geophys[J].Expanded Abstracts,2003,153～156．
[7] Zhou H B,Zhang G Q,Bloor R,An anisotropic acoustic wave equation for modeling and migration in 2D TTI media.76th Ann.Internat.Mtg.,Soc.Expl.Geophys[J].Expanded Abstracts,2006,194～197．
[8] Dewangan P,Tsvankin I.Modeling and inversion of PS-wave moveout asymmetry for tilted TI media:Part Ⅰ-Horizontal TTI layer[J].Geophysics,2006,71(4) :D107～D122．
[9] Dewangan P,Tsvankin I.Modeling and inversion of PS-wave moveout asymmetry for tilted TI media:Part 2 Dipping TTI layer[J].Geophysics,2006,71(4) :D123～D134．
[10] 牛滨华,孙春岩.半空间均匀各向同性单相固体弹性介质与地震波传播[M].北京:地质出版社,2006． Niu B H,Sun C Y.Half-space homogeneous isotropic single phase solid elastic medium and seismic wave propagation[M].Beijing:Geological Publishing House,2006．
[11] 邓继新,史謌,刘瑞珣,等.泥岩、页岩声速各向异性及其影响因素分析[J].地球物理学报,2004,4 7(5) :862～868． Deng J X,Shi G,Liu R X,et al.Analysis of the velocity anisotropy and its affection factors in shale and mudstone[J].ChineseJ.Geophys.(in Chinese),2004,47(5) :862～868．
[12] 郑海山,张中杰.横向各向同性(VTI)介质中非线性地震波场模拟[J].地球物理学报,2005,48(3) :660～667． Zheng H S,Zhang Z J.Synthetic seismograms of nonlinear seismicwaves in anisotropic (VTI) media [J].Chinese J.Geophys.(in Chinese),2005,48(3) :660～671．
[13] 杜向东,翁斌,刘军荣,等.TI介质偏移速度建模研究[J].地球物理学报,2008,51(2) :538～545． Du X D,Weng B X.Liu J R,et al.Migration velocity modeling strategies of TI media[J].Chinese J.Geophys.(in Chinese),2008,51(2) :538～545．
[14] 张家芳,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 homogenous background velocity model [J].Chinese J.Geophys.(inChinese),2008,51(4) :1180～11187．
[15] 黄超,董良国.可变网格与局部时间步长的高阶差分地震波数值模拟[J].地球物理学报,2009,52(1) :1 76～186． Huang C,Dong L G.High-order finite-difference method in seismic wave simulation with variable grids and local timesteps[J].Chinese J.Geophys.(in Chinese),2009,52(1) :176～186．
[16] 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,etal.2． 5D Numerical simulation of seismic wave in inhomogeneous media [J].Progress in Geophysics,2008,23(4) :1092～1098．
[17] 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 modeling with quasi isotropic approximation in weakly anisotropic mediurn[J].Progress in Geophysics,2008,23(4) :1118～1123．
[18] 牛滨华,何樵登.孙春岩.六方各向异性介质方位矢量波动方程及其相速度[J].石油物探,1 994,33(1) :19～29． Niu B H,He Q D,Sun C Y.Azimuth vector wave equation and its phase velocity in hexagonal anisotropy medium[J].GPP,1994,33(1) :19～29．
[19] 郝重涛,姚陈,王迅.任意空间取向TI介质中速度随方位变化特征[J].地球物理学进展,2006,21(2) :524～530． Hao C T,Yao C,Wang X.The characteristics of velocities with azimuth variation for arbitrary spatial orientation TI media[J].Progress in Geophysics,2006,21(2) :524～530．
[20] 郝重涛,姚陈.任意空间取向TI介质中体波速度特征[J]. 地球物理学报,2007,50(2) :546～555． Hao C T,Yao C.Analysis of bodywave velocity characteristic for TI medium with arbitrary spatial orientation [J].ChineseJ.Geophys.(in Chinese),2007,50(2) :546～555．
[21] 赵爱华,丁志峰.一种弱各向异性介质地震波群速度的近似表示新方法[J].地球物理学进展,2005,20(4) :916～919． Zhao A H,Ding Z F.Ne w approximate expressions of Seismic group velocities for weakly anisotropic media [J].Progress in Geophysics,2005,20(4) :916～919．
[22] Xuan Y H,He Q D,Lin Y.Phase velocity in 2D TTI media [J].Applied Geophysics,2007,4(1) :25～28．
[23] Winterstein D F.Velocity anisotropy terminology for geophysicists[J].Geophysics,55(8) :1070～1088．
[24] Berryman J G.Long-wave elastic anisotropy in transversely isotropic media[J].Geophysics,1979,44(5) :896～917．
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