Lowrank one-step wave extrapolation for reverse-time migration

Discussion

The modeling experiments using an isotropic two-layer model and a portion of BP 2007 TTI benchmark model show that the proposed operator is in practice remarkably stable and accurate. When the velocity contrast is not very sharp, the lowrank one-step method is capable of propagating waves free of dispersion artifacts using time steps even larger than the Nyquist sampling limit of the source wavelet. The maximum efficiency can be achieved using the largest step size possible, which in RTM applications usually corresponds to the time sampling of the imaging condition. In contrast, conventional approaches, such as high-order finite-difference and pseudo-spectral methods, are confined to small step sizes due to severe stability and accuracy constraints. This is especially true when S-wave extrapolation is required by RTM, because the slow S-wave velocity at shallow depths imposes a strict constraint on the time step size of conventional methods. Lowrank one-step wave extrapolation has been recently applied to converted wave imaging by Casasanta et al. (2015) and achieved accurate results.

Our numerical experiments confirm advantages of the lowrank one-step wave extrapolation over the two-step scheme. By extrapolating an analytical wavefield with the imaginary part related to the first derivative of the real-valued wavefield, the one-step scheme is capable of using a much larger time step size (Du et al., 2014). The complex phase function used by the lowrank one-step method offers additional freedom in the design of the wave extrapolation operator. In media with smoothly varying velocities, a large time step may impair accuracy using the conventional formulation. By using a more accurate expression, for example by admitting more terms from the Taylor expansion of the phase function, the lowrank one-step wave extrapolation can achieve higher accuracy. The complex-valued extrapolation operator also allows for an effective mixed-domain absorbing boundary condition, which dampens wave energy according to the phase direction and thus avoids artificial reflections at large incident angles. The application of a complex phase function is not limited to the proposed cases. Another possible extension is seismic modeling and imaging in visco-acoustic media (Zhu and Harris, 2014), where the one-step extrapolator solves a complex-valued, decoupled dispersion relation that incorporates attenuation effects (Sun et al., 2014,2015).

Lowrank one-step wave extrapolation for reverse-time migration