Least-squares RTM in nonorthogonal coordinates and applications to VTI media

Applying reverse time migration (RTM) to seismic data often results in wavefield propagation fraught with migration artifacts. To overcome this, we introduce least-squares RTM (LSRTM), which is applied to the migrated section via the Born approximation and the conjugate gradient algorithm. LSRTM extrapolates the reconstructed wavefield using a wave equation that has been transformed into the Riemannian domain. This approach addresses the oversampling effect of seismic signals by ensuring even sampling and allows for the recovery of greater amplitude in the final migrated image. For each point in the Cartesian coordinate system, there is a corresponding vertical time point. Consequently, we can interpolate the reconstructed source wavefield in the new ray coordinates by drawing a Cartesian–Riemannian mapping function. The specific finite difference (FD) scheme and boundary conditions notwithstanding, the Riemannian wavefield extrapolator operates via two formulas depending on the type of wave equation used. In vertical transversely isotropic (VTI) media, velocity tends to decrease with depth, significantly distorting the migration results. This issue can be resolved by applying the LSRTM in either the Cartesian or pseudodepth domain, supported by a proper wavefield extrapolator. The finite-difference Riemannian wavefield extrapolator, when applied to the Born modeled seismic data, produces results strikingly similar to the classical LSRTM, albeit with some amplitude differences owing to various implementation issues and the oversampling effect. Our results strongly indicate that the domain transformation strategy effectively reduces computational time without compromising the accuracy of the Cartesian-mesh-typed LSRTM results.