High-order kernels for Riemannian Wavefield Extrapolation

Extrapolation kernels

Extrapolation using equation (13) implies that the coefficients defining the problem, $a$ and $b$, are not changing spatially. In this case, we can perform extrapolation using a simple phase-shift operation

$$\mathcal{U}_{{ \tau}+\Delta{ \tau}} = \mathcal{U}_{{ \tau}} e^{i k_\tau \Delta{ \tau}} \;,$$ (15)

where $\mathcal{U}_{\tau +\Delta\tau }$ and $\mathcal{U}_{\tau }$ represent the acoustic wavefield at two successive extrapolation steps, and $k_\tau$ is the extrapolation wavenumber defined by equation (13).

For media with lateral variability of the coefficients $a$ and $b$, due to either velocity variation or focusing/defocusing of the coordinate system, we cannot use in extrapolation the wavenumber computed directly using equation (13). Like for the case of extrapolation in Cartesian coordinates, we need to approximate the wavenumber $k_\tau$ using expansions relative to $a$ and $b$. Such approximations can be implemented in the space-domain, in the Fourier domain or in mixed space-Fourier domains.