Decoupled wave equation and forward modeling of qP wave in VTI media with the new acoustic approximation

Acoustic approximation has received wide attention in modeling and inversion for the anisotropic wave equation to avoid high computational cost and parameter trade-off in seismic inversion. However, it also is limited by the stability condition, such as the instability for the qP-wave equation in transversely isotropic media with $ε<δ$⁠. We have developed a new approach that decoupled wave equation and forward modeling of the qP wave in vertical transversely isotropic (VTI) media with the new acoustic approximation. To keep dispersion relations (⁠$ωSV$⁠) of the qSV wave in each direction equal to zero, we formulate the vertical S-wave velocity (⁠$VS0$⁠) to be a function of the model parameters and wavenumber component (⁠$kx,kz$⁠), rather than setting it to zero. Then, the corresponding dispersion relation of pure qP for the new acoustic approximation is derived. According to this decoupled dispersion relation, we obtain the decoupled wave equation of pure qP wave in VTI media by inverse Fourier transform. To solve the wave equation in the space domain efficiently, the operator $Sk$ in the wave equation is characterized in the space domain by its asymptotic approximation operator $Sn$⁠. From the qP wave equation in the time-space domain, we realize the forward modeling of the pure qP wave in VTI media with the finite-difference method. The dispersion relation analysis and numerical examples find that the decoupled qP-wave equation with the new acoustic approximation does not contain the degenerate qSV wave and is a pure qP-wave equation, which is in good agreement with the simulation results of the elastic wave equation and has high accuracy and is stable in VTI media with $ε≥δ$ or $ε<δ$⁠.