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

In this paper, we derive the viscoacoustic and viscoelastic velocity-stress wave equations in constant-Q attenuation 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.