Conclusions

We extend one-way wavefield extrapolation to Riemannian spaces which are, by definition, described by non-orthogonal curvilinear coordinate systems. We choose semi-orthogonal Riemannian coordinates which include, but are not limited to, ray coordinate systems.

We define an acoustic wave-equation for semi-orthogonal Riemannian coordinates, from which we derive a one-way wavefield extrapolation equation. We use ray coordinates initiated either from a point source, or from an incident plane wave at the surface. Many other types of coordinates are acceptable, as long as they fulfill the semi-orthogonal condition of our acoustic wave equation.

Since wavefield propagation happens mostly along the extrapolation direction, we can use cheap $15^\circ$ finite-difference or mixed-domain extrapolators to achieve high angle accuracy. If the ray coordinate system overturns, our method can be used to image overturning waves with one-way wavefield extrapolation.

A special case of extrapolation corresponds to coordinates initiated by a plane wave at the surface propagating initially in the vertical direction. Since our extrapolation is done as a function of one-way traveltime, this case resembles imaging in vertical traveltime, although it is more physically correct, since it allows lateral movement of energy, which is not the case for vertical $\tau$ imaging.

Two main applications of our method are imaging of steeply dipping or overturning reflections.