Least-squares reverse-time migration using one-step two-way wave extrapolation by non-stationary phase shift

The one-step low-rank wave-extrapolation operator is a nonstationary combination filter, and, as such, corresponds to the well-known PSPI method for the one-way wave equation. We propose its adjoint operator, the non-stationary phase shift (NSPS) wave extrapolation in time, which is a non-stationary convolution filter. A combination of PSPI and NSPS methods creates a symmetric and more accurate wave propagator. Numerical examples using a two-layer model demonstrate the improved accuracy. For applications to prestack reverse-time migration (RTM), the proposed framework incorporates a complex valued imaging condition. We use the forward modeling operator and its adjoint to implement least-squares RTM (LSRTM). Synthetic examples illustrate that, compared with conventional RTM, the low-rank LSRTM is capable of suppressing migration artifacts and achieving better illumination.