Conventional modeling and imaging for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear artifacts due to the coupling of the P- and SV-wave modes in the coupled equations. On the contrary, the decoupled equations for TTI media provide a more stable solution due to the separated P- and S-wave modes. Because the decoupled equations involve complicated pseudo-differential operators in space, it is more convenient to apply the pseudo-spectral method to these equations. However, the second-order time-stepping scheme of the pseudo-spectral method may suffer from time-stepping errors and instabilities for a large time step. We have developed an optimized pseudo-differential operator (OPO) that incorporates not only the spectral evaluation of the pseudo-differential operator but also a temporal correction term that would effectively compensate the time-stepping errors associated with the time-wavenumber domain extrapolation of the wave equation. The OPO was constructed through multiplying the symbol of the OPO by the normalized pseudo-Laplacian operator, which contains a variable compensation velocity. It was efficiently solved through the low-rank decomposition. We have applied OPOs to solve the TTI decoupled equation to simulate the pure acoustic wave. Our 2D and 3D synthetic results demonstrate that the proposed method has high accuracy in time and space with relaxed stability conditions compared with the conventional pseudo-spectral method. The low rank of symbols of OPOs makes the proposed method more efficient than the dispersion relation-based low-rank wave extrapolation and pseudo-analytical methods.

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