TTI介质弹性波近似解耦波动方程

Approximate decoupled wave equations for elastic waves in TTI media

借助Christoffel方程可求解出各向异性介质弹性波精确频散关系.利用近似方法进行处理,再通过傅里叶逆变换将频率波数域算子变换为时空域算子,可导出解耦的qP波或qS波波动方程.本文在TTI介质弹性波精确频散关系的基础上,利用近似配方法推导了qP波和qSV波近似频散关系,通过傅里叶逆变换推导了TTI介质qP波和qSV波解耦的波动方程.为了验证近似频散关系的有效性,利用两组模型参数对其进行数值计算,分析了相对误差在不同传播方向上的分布.随后使用有限差分方法分别对均匀、层状及复杂TTI介质弹性波近似解耦波动方程进行数值模拟,结果显示qP波和qSV波完全解耦,并且在各向异性参数η<0以及介质对称轴倾角变化较大的情况下,纯qP波和纯qSV波近似波动方程依然可以保持稳定.

The exact dispersion relation equation of elastic waves can be derived by solving the Christoffel equation in anisotropic media.Using approximate methods,and then converting to the frequency-wavenumber domain operator to time-space domain operator by using inverse Fourier transform,the decoupled qP wave or qSV wave equation can be derived.The approximate method of completing the square is adopted to derive the approximate wave equation of qP and qSV waves based on the exact dispersion relation of elastic waves in TTI media,then the approximate qP and qSV wave equations can be derived by using inverse Fourier transform.In order to verify the validity of the approximate dispersion relation equation,two sets of models are tested on numerical simulation,and the distribution of relative error in different propagation directions is analyzed.Subsequently,the finite-difference method is imployed to simulate the wave equations of the elastic wave for the TTI models in homogeneous,layered and complex media models,respectively.Results show that qP and qSV waves are completely decoupled,and the approximate wave equations of pure qP and pure qSV waves remain stable in the media withη<0 parameters and highly varying anisotropy angels.