The Characteristics Of Seismic Wave Propagation And The Numerical Simulations Of Wave Equations In TI Media

The underground media are generally velocity anisotropy,which means that the seismic wave speed is determinated by the direction of wave propagtion.Sedimentary strata usually show transversely isotropy(TI media).Acorrding to the spatial direction of symmetry axis,TI media are classified into four types,namely vertical transversely isotropic media,titled transversely media,horizontal transversely media and TI media with arbitrary spatial orientation.Rotating the VTI media along the horizontal Y axis leads to TTI media.Similarlily,rotating the TTI media along the vertical Z axis,we can obtain the ATI media.HTI media are the special cases of ATI media with horizontal symmetry axis.Generally,the method to calculate the phase velocity of three-dimensional TTI or ATI media is as following: The stiffness matrix of TTI and ATI media is obtained by bond transformation of VTI media stiffness matrix.Then the stiffness matrix of TTI or ATI media is substituted into Christoffel equation.By calculating the eigenvalue of Christoffel matrix,the phase velocity of TTI and ATI media can be obtained.It is difficult to derive the group velocity expressed in group angle.Due to the fact that group velocity depends on phase velocity,we compute group velocity based on phase velocity.The method of calculating phase velocity and group velocity by Bond transformation is very complex.Based on the phase velocity and group velocity of VTI media,we present an approach to calculate the phase velocity and group velocity in three-dimensional TTI and ATI media by using rotated vector.This method has a clear geometric meaning and decrease the amount of calculation.The direction of phase velocity is the same as that of propagation vector.The module of phase velocity of TTI and ATI media can be solved by calculating the product of unit vector and scalar product of propagation vector in the direction of symmetry axis of TI media,and then the vector form of phase velocity can be obtained by multiplying with propagation vector.First,we need to project the propagation vector of observer coordinate into the constitutive coordinate system,and then obtain the phase velocity and group velocity in the constitutive coordinate system.Finally,rotate the phase velocity and group velocity vector in the constitutive coordinate system in reverse direction to obtain the phase velocity and group velocity of ATI medium in the observed coordinate system.The exact analytical expressions of phase velocity,group velocity and polarization vector of seismic wave in TI media are very complex.It is found that the sedimentary strata near the surface show weak anisotropy.With the help of Thomsen anisotropic parameters,a series of approximate formulas can be derived,which simplifies the problem.Based on Thomsen's weak anisotropy phase velocity formula and Crampin's group velocity theory of anisotropic media,the group velocity and polarization vector of two-dimensional VTI media are derived.The vertical plane in any direction of the three-dimensional VTI medium can be regarded as a two-dimensional VTI medium.A local coordinate system is constructed which changes with the propagation azimuth.The unit vector(propagation vector)in the propagation direction is orthogonal to a horizontal coordinate axis of the local coordinate system.Using the local coordinate system,we can easily extend the group velocity and polarization vector of two-dimensional VTI media to three-dimensional VTI media.Then,using the method of vector rotation,we derive the analytical expressions of the group velocity and the polarization vector of three-dimensional TTI,ATI media.The polarization directions of qP wave,qSV wave and SH wave in TI medium are perpendicular to each other,and the normalized polarization vector of SH wave is obtained by calculating the cross multiplication of the vector of symmetrical axis and the vector of propagation.The polarization vector of qSV wave is obtained by cross multiplication of the polarization vector of SH wave and qP wave.Based on the phase velocity of seismic wave in TTI(ATI)medium,the decoupled qP-qSV wave equation is derived by using Taylor expansion formula.The stability condition of the wave equation is analyzed.The error analysis shows that the accuracy of the second-order approximate equation is higher than that of the first-order approximate equation.The numerical results of pseudo spectral method show that the second-order approximate decoupled qP-qSV wave equation is still stable in complex media,and separates the qP wave and qSV wave completely.