New Acoustic Pure P- and S-Wave Equations for Modelling and Reverse Time Migration in Tilted Transversely Isotropic Media

Abstract

To reduce information loss in seismic imaging of complex geological media, it is essential to account for anisotropy, particularly in tilted transversely isotropic (TTI) formations–the most common anisotropy in exploration geophysics. However, the elastic wave equations governing such media inherently couple $P$- and $S$-waves, whereas acoustic reverse time migration (RTM) relies on pure $P$-wavefields. This coupling introduces shear-wave energy during modelling, leading to artefacts in the migrated image. To address this issue, exact acoustic pure $P$- and $S$-wave equations are derived from the analytical decomposition of the elastic wave equation in vertically transversely isotropic media. Their dispersion relations are mathematically equivalent to those of the exact elastic wave equation, and a simple yet effective optimization procedure is proposed for their numerical solution in TTI media. The proposed formulations accurately decouple the $P$- and $S$-wavefields with near-zero error in both the phase-angle and spatial domains – where, in the latter, the error is also near zero in both phase and amplitude – even when the $S_V$-wave velocity is nonzero and $\delta >\epsilon$. Moreover, the pure $P$-wave formulation is shown to be efficient and stable for seismic modelling and RTM in complex anisotropic media, as demonstrated by synthetic TTI modelling and migration experiments using the pseudo-spectral method.