Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation (Communications in Computational Physics, 2012)
We develop a stable finite difference approximation of the three-dimensionalviscoelastic wave equation. The material model is a super-imposition of N standardlinear solid mechanisms, which commonly is used in seismology to model a materialwith constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, makingit significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finitedifference scheme for the elastic wave equation in second order formulation [SIAM J.Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposeddiscretization is energy stable, even in the case of variable material properties. Theproof relies on the summation-by-parts property of the discretization. The new schemeis implemented with grid refinement with hanging nodes on the interface. Numericalexperiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are usedto demonstrate how the number of viscoelastic mechanisms and the grid resolutioninfluence the accuracy. We find that three standard linear solid mechanisms usuallyare sufficient to make the modeling error smaller than the discretization error.