任意空间取向TI介质qP和qSV波解耦的波动方程

A decoupled qP-wave and qSV-wave equation in TI media with arbitrary spatial oriented symmetry axis

由于构造运动等作用,TI介质对称轴往往沿空间任意方向分布,具有任意空间取向对称轴的TI(ATI)介质更符合实际地质情况.VTI介质与ATI介质的相速度在形式上具有一致性,VTI介质中地震波的相角对应ATI介质对称轴与地震波传播方向的夹角.本文基于Tsvankin的VTI介质精确相速度公式,利用TI介质对称轴和地震波传播方向上单位向量的数量积和向量积来计算ATI介质的精确相速度.根据弱各向异性假设,导出qP波和qSV波的近似相速度,分析了近似公式的误差,讨论总结了ATI介质qP波和qSV波的相速度特征.本文中的单位向量采用观测坐标系表示,通过相角关系,可以较为方便地由ATI介质近似相速度导出频散关系,然后借助傅里叶逆变换推导出时间-波数域qP波和qSV波解耦的波动方程.数值算例表明本文的波动方程是qP波和qSV波解耦的,波场计算结果稳定,未出现明显的数值频散,验证了本文方法的有效性.

Due to the tectonic deformation and other geological factors, the symmetry axis direction of transversely isotropic media is arbitrary. Formally, the phase angle of plane waves travelling in VTI (transversely isotropic media with a vertical symmetry axis) media is consistent with the angle between the symmetry axis and the direction of plane waves travelling in ATI media. With the accurate phase velocity derived by Tsvankin in VTI media, we calculate the accurate phase velocity of quasi-P and quasi-SV waves for ATI media (transversely isotropic media with arbitrary spatial orientation symmetry axis) in terms of the dot product and cross product of the unit vectors of the symmetry axis and the direction of wave propagation. According to the weak anisotropy assumption and Taylor′s formula, we derive the approximate formulas of phase velocity of qP- and qSV-waves. The errors of the approximate formulas are analyzed. The phase velocity characteristics of qP-wave and qSV-wave in ATI media are discussed. The unit vectors are represented by the observational coordinate system. Based on the approximate phase velocity formulas of ATI media, the decoupled qP and qSV-wave equations in the time-wavenumber domain are derived with the phase angle relations by using inverse Fourier transform. Numerical examples demonstrate that the qP and qSV wavefields are decoupled and stable without obvious numerical dispersion, showing the validity of the method proposed in this study.