Dispersion and Stability Condition of Seismic Wave Simulation in TTI Media

Abstract

For seismic waveform simulation in tilted transversely isotropic (TTI) media, we derive explicitly the numerical dispersion relation and the stability condition for the computation of a 2D pseudo-acoustic wave equation. The numerical dispersion relation indicates that the number of sampling points per wavelength has the greatest influence on the dispersion, while the anisotropic parameters of the TTI media and the mesh rotation angle have little influence on the dispersion. Given an appropriate spatial sampling, the stability condition is for the selection of the time step for the implementation of the TTI wave equation. We partition a numerical model using quadrangle grids in Cartesian coordinates, and map it to a computing model in which any nonrectangular meshes in Cartesian coordinates become rectangular meshes. Then we reformulate the pseudo-acoustic wave equation for the TTI media accordingly in the computational space. We implement seismic waveform simulation using the second-order finite-difference method straightforwardly, and show examples with a desirable accuracy using a model with non-rectangular meshes in Cartesian coordinates along a curved surface and fluctuating interfaces in the TTI media.

About partial boundary dissipation to Timoshenko system with delay

“…To numerical results that can be applied to this model we suggest Rao and Wang. 19 The delay was not considered in the above references. The main result of this paper is to prove the exponential stability to partial boundary dissipation with delay, provided the wave speed are equals and Equation (3.17) holds.…”

“…Instead in Munoz Rivera and Naso 18 under the same condition (), the authors proved the exponential stability. To numerical results that can be applied to this model we suggest Rao and Wang 19 . The delay was not considered in the above references.…”