三维TTI介质中的弹性波场分解

Elastic wavefield decomposition in 3D TTI media

根据三维TTI介质倾斜对称轴与观测坐标系之间的几何关系,经坐标旋转,可以将其近似看成三维VTI介质,故适用于VTI介质的波场分解方法就可以应用于TTI介质。为此,从三维各向异性弹性波动方程出发,通过求解Christoffel方程的特征向量,构建了三维VTI介质的波场分解算子;再利用坐标旋转推导了三维TTI介质的波场分解算子;最后通过泊松方程实现三维TTI介质的弹性波场分解。引入改进的快速算法,避免了直接求解泊松方程。推导的分解算子考虑了随空间变化的弹性参数和倾斜对称轴,可适用于复杂的三维TTI介质模型。当各向异性参数以及对称轴倾角都为零时,该算子可以退化为各向同性的形式。数值计算结果表明,该TTI介质波场分解算子比各向同性或VTI介质算子能得到更精确的分离结果,可高效实现三维各向异性矢量弹性波场分解。

According to the geometric relationship between the tilted symmetry axis of 3D TTI media and the observation coordinate system,the TTI media can be regarded as 3D VTI media through coordinate rotation.The wavefield decomposition method for VTI can be applied to TTI media.Therefore,on the basis of the 3D anisotropic elastic wave equation,this paper built a 3D VTI wavefield decomposition operator by solving the eigenvectors of the Christoffel equation.Then a 3D TTI decomposition operator is derived by coordinate rotation.Finally,the elastic wavefield decomposition of 3D TTI can be realized by Poisson’s equation.An improved fast algorithm is introduced to avoid solving Poisson’s equation directly.The derived decomposition operator is suitable for the complex 3D TTI media because it considers the varying elastic parameters and tilted symmetry axis with space.When the anisotropic parameters and tilted angle of the symmetry axis are zero,this operator can be degraded into isotropic form.Numerical results show the derived TTI operator can obtain more accurate decomposition results than isotropic or VTI operators,and efficiently realize 3D anisotropic vector elastic wave field decomposition.