Reverse-time Migration in Tilted Transversely Isotropic Media with Decoupled Equations

摘要

Conventional modeling and migration for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear wave artifacts due to the coupling of the P-wave and SV-wave modes in the TTI coupled equations. Starting with the separated P- and SV-phase velocity expressions for vertical transversely isotropic (VTI) media, I extend these decoupled equations for modeling and reverse-time migration (RTM) in acoustic TTI media. Compared with the TTI coupled equations published in the geophysical literature, the new TTI decoupled equations provide a more stable solution due to the complete separation of the P-wave and SV-wave modes. The pseudospectral (PS) method is the most convenient method to implement these equations due to the form of wavenumber expressions and has the added benefit of being highly accurate and thus avoiding numerical dispersion. The rapid expansion method (REM) in time is employed to produce a broad band numerically stable time evolution of the wavefields. Synthetic results validate the proposed TTI decoupled equations and show that modeling and RTM in TTI media with the decoupled P-wave equation remain numerically stable even for models with strong anisotropy and sharp contrasts.The most desirable feature of the TTI decoupled P-wave equation is that it is absolutely free of shear-wave artifacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward-backward Fourier transforms in wavefield extrapolation at each time step, the computational cost is also high, and thereby hampers its prevalence. I hereby propose to use a hybrid pseudospectral and finite-difference (FD) scheme to solve the TTI decoupled P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D RTM examples using the hybrid solution to the decoupled P-wave equation are carried out, and respective runtimes are listed and compared. Computation examples show that the hybrid strategy demands less computation time and is faster than using the pseudospectral method alone. Furthermore, this new hybrid TTI RTM algorithm is less computationally expensive than the FD solution to the conventional TTI coupled equations but more stable.