常Q衰减介质分数阶波动方程优化有限差分模拟

Modeling of wave equation with fractional derivative using optimal finite-difference method in constant-Q attenuation media

本文基于Kjartansson常Q模型理论,推导了常Q衰减介质中黏声波和黏弹性波的速度-应力方程,并采用基于二项式窗函数的优化交错网格有限差分方法进行了数值模拟,同时引入不分裂的复频移卷积完全匹配层(CPML)吸收边界条件,以消除边界反射.使用基于自适应时间步长记忆方法的中心差分近似时间分数阶导数,与常用的短时记忆方法相比,提高了波动方程的离散化精度和计算效率.通过对比均匀模型下声波的数值解与解析解,验证了算法的精确性,并进一步分析了不同品质因子下地震波的频散及衰减特征.对BP盐丘模型的数值模拟结果可以较好地反映本文数值方法对复杂介质的适应性及频散压制效果.

In this paper,we derive the viscoacoustic and viscoelastic velocitystress wave equations in constant-Qattenuation medium.The optimal staggered grid finite-difference method based on binomial windows are used to numerically solve the equations,with incorporating convolutional perfectly matched layer（CPML）boundary conditions to eliminate boundary reflections.We introduce the adaptive time step memory method to approximate the time fractional derivative,which improves the discretization accuracy and computational efficiency of the wave equation comparing with the short memory method.Furthermore,we evaluate the accuracy of the algorithm by comparing the numerical solution with the analytic solution of the acoustic wave for the homogeneous media,and further analyze the dispersion and attenuation characteristics of seismic wave under different quality factors.Finally,we consider the BP salt model to demonstrate the applicability of our numerical algorithm in heterogeneous medium with obvious effect in suppressing numerical dispersion.