Conclusions

This chapter extends one-way wavefield extrapolation to Riemannian spaces which are described by non-orthogonal curvilinear coordinate systems. I use semi-orthogonal Riemannian coordinates that include, but are not limited to, ray coordinate systems.

I define an acoustic wave-equation for semi-orthogonal Riemannian coordinates, from which I derive a one-way wavefield extrapolation equation. I use ray coordinates that can be 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 the acoustic wave equation in Riemannian coordinates.

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

Riemannian coordinates are better suited for wavefield extrapolation, because they do not restrict wave propagation to the preferential vertical direction but allow numerical wave extrapolation to follow the direction of the natural wave propagation. Coordinate system triplications pose challenges that can be solved numerically, but which are better avoided.