Stability

Separation of the P wave and SV wave modes makes the new TTI decoupled equations more stable than conventional TTI coupled equations. However, in practice, a potential instability problem still remains and appears as``ringing'' artifacts in the wavefield. The presence of mixed spatial derivatives and strong contrasts in the anisotropic parameters, particularly the axis of symmetry, seems to trigger the problem. And this instability problem is restricted to wavenumbers near Nyquist. To deal with these high frequency spatial oscillations and thus stabilize the computation, a band pass filter is introduced in my implementation and applied to the wavenumber operators that smoothly tapers them to zero at the Nyquist wavenumber. A Hanning taper, Zhou and Greenhalgh (1992), is used staring at 90% of the Nyquist wavenumber and having a value of zero at the Nyquist. My experiences show that such a filter with a window length of 10 percent of the band perfectly removes the "ringing" artifacts while preserving the accuracy to the highest extent. By doing so, there is no need to compute mixed spatial derivatives on staggered grids (Corrêa et al., 2002) and afterward interpolate back to normal grids.