TTI介质准P波和准SV波方程的迭代有限差分解法

An iterative finite-difference solution for quasi-P and quasi-SV wave equations in TTI media

本文介绍了时间和空间微分耦合的具有倾斜对称轴的横向各向同性(TTI)介质准P波和准SV波方程,并把该方程从时间—波数域转换到时间—空间域,提出了该方程的两种有限差分解法并给出稳定性条件。解法一按常规有限差分近似式对波动方程进行逼近,每个时间步波场的计算都需要求解一个大型稀疏矩阵线性方程组;解法二先对各网格节点的波场赋以零值,再通过由有限差分近似建立的公式逐个节点进行迭代计算,每次计算仅使用少数网格节点的波场值,不需要求解方程组。与伪谱法的理论模型计算结果对比表明,迭代有限差分解法是正确的,虽然其计算精度较低,但计算速度较快,而且具有较好的收敛性。

In this work,we transform the available temporal and spatial derivatives coupled quasi-P and quasi-SV wave equations for transversely isotropic media with a tilted symmetric axis(TTI) media from time-space domain to time-frequency domain.We present two approaches to calculate their solutions by finite-difference(FD) approximation and provide the stability conditions.While one approach computes the wave fields of each time step by solving a large sparse linear system based on FD approximation of the wave equations,the other approach computes the wave fields through an iterative algorithm built by FD approximation after assigning initial zero wavefield values,on the basis of the former approach,and thus no need to solve the large linear system.The iterative computation for each grid node uses only a small number of node values.Comparisons with the model test results of pseudo-spectral approach demonstrate the presented iterative FD approach is correct and is more efficient but less accurate.Analysis to the variation curves between updating value and iteration times shows the convergence of the iterative algorithm.