Hybrid PS/FD numerical simulation and stability analysis of pure P-wave propagation in VTI media

Seismic anisotropy is undergoing rapid expansion as a result of recent advances in parameter inversion and seismic processing. The simplest and most commonly used anisotropic model is the transverse isotropy (TI) with a vertical symmetry axis (VTI) or with a tilted symmetry axis (TTI). TI models can provide good representation of intrinsic anisotropy of the earth media. Vertical and tilted transverse isotropy have become an integral part of velocity fields in prestack depth migration algorithms for imaging complicated anisotropic structures, especially those algorithms based on the wave equation such as reverse time migration (RTM). Reverse time migration in anisotropic media based on the full elastic anisotropic wave equation demands tremendous computation and requires an anisotropic model that simultaneously contains P- and S-wave velocity information. However, the seismic data recorded with conventional vertical geophones often mainly contain P-wave reflections. So an alternative Way to reduce parameter requirement and computational cost for P-wave modeling and reverse time migration in TI media is to apply an acoustic approximation. Acoustic approximation yields a kinematically accurate P-wave propagation for TI media, however, this method can't completely get rid of shear-wave noise. Moreover, acoustic approximation equations become unstable for anisotropic models with parameters epsilon < delta and for heterogeneous models with highly varying dip and azimuth angles. For these above mentioned reasons, the pure P-wave equation for modeling and migration in TI media has attracted more and more attention. The pure P-wave equations are also called decoupled equations which are derived from the exact dispersion by using the first-order Taylor series expansions. Here, we propose the first-order form of the pure P-wave equation with stress and particle velocities as wavefield variables for VTI media. Our main work is restricted to pure P-wave mode for practicality. Just like most other anisotropic pure wave mode equations, the proposed first-order pure P-wave equation also involves pseudo-differential operators in space which are difficult to handle by using finite-difference (FD) method directly. The most natural choice to implement these equations is pseudospectral (PS) method. Although pseudospectral method requires only two nodes per wavelengths, it is still computationally expensive due to several forward-backward Fourier transforms in wavefield extrapolation at each time step. Here we adopt a hybrid pseudospectral and finite-difference scheme to solve the first-order pure P-wave equation. The hybrid PS/FD scheme requires only two Fourier transforms at each time step in 2D case and thus it is faster than the pseudospectral method alone. In addition, pure P-wave equation is completely free from shear-wave, so one can use large discretization step in space to reduce the computational cost further. The stability of the hybrid PS/FD scheme on regular and staggered grids is investigated by von Neumann's method. The analysis shows that the staggered-grid hybrid PS/FD scheme is more stable than regular-grid hybrid PS/FD scheme. The hybrid PS/FD scheme on regular grid is unstable when the anisotropy parameteris a greater than a and this instability problem is restricted to wavenumbers near Nyquist. To deal with these high frequency spatial instabilities, a low pass filter is applied to the wavenumber operators in our implementation. We test the hybrid PS/FD scheme with various kinds of VTI model. The numerical simulations provide insight into the kinematic and dynamic accuracy of the pure P-wave VTI equation in comparison with the elastic wave equation. The numerical examples testify that: (1) Pure P-wave equation is completely free from shear-wave compared to elastic wave equation and conventional acoustic VTI equation; (2) The proposed pure P-wave equation can provide good kinematic and dynamic approximations to the elastic wave equation for homogeneous media with weak-to-moderate anisotropy and is more efficient than full elastic wave equation; (3) The hybrid PS/FD scheme on regular grid suffers from some numerical instabilities when the anisotropy parameteris a greater than and the low pass filtering approach do improve the stability; (4) The hybrid PS/FD scheme on staggered grid for pure P-wave equation has better stability than the regular-grid PS/FD scheme. Even in complex heterogeneous media, it still can provide stable and accurate modeling of P-wave propagation. The numerical examples demonstrate the hybrid PS/FD scheme is an effective and efficient method to solve the first-order velocity-stress pure P-wave equation. At the same time, it should be noted that the staggered-grid hybrid PS/FD scheme is more numerically stable than regular-grid hybrid PS/FD scheme, however, its stability condition still is stricter than that of acoustic equation.