Riemannian wavefield extrapolation is a method for propagating wavefields on generalized coordinate meshes. The central idea behind RWE is transforming the geometry of the full domain to one where the extrapolation axis is oriented in the general wavefield propagation direction. Solving the corresponding one-way extrapolation equations propagates the bulk of wavefield energy at angles relatively close to the extrapolation axis, thus improving wavefield extrapolation accuracy. One obvious application is generating high-quality Green's functions for point-sources, where a ray-based coordinate system is first generated by ray-tracing through the velocity model and then used as the skeleton on which to propagate wavefields.
Although the full-domain decomposition approach naturally adapts to propagation in a point-source ray-coordinate system, two unresolved issues make it difficult to apply RWE efficiently in the prestack domain. First, receiver wavefields are usually broadband in plane-wave dip spectrum and cannot be easily represented by a single coordinate system (i.e. opposing dips propagate in opposing directions). Second, optimal source and receiver meshes usually do not share a common geometry. This factor is detrimental to algorithmic efficiency where generating images by correlating source and receiver wavefields: by existing on different grids they must both be interpolated to a common Cartesian reference frame prior to imaging. This leads to a significant number of interpolations that leaves the algorithm computationally unattractive.
The main goal of this paper is to specify a single coordinate system that enables the accurate propagation of high-angle and overturning components of both the source and receiver wavefields. To these ends, we demonstrate that an elliptic coordinate system is a ``natural'' prestack shot-profile migration coordinate system exhibiting nice geometric properties. An elliptic coordinate system originates on the surficial plane and steps outward as a series of ellipses. Thus, the coordinate system expands in a radial-like manner appropriate for computing accurate point-source Green's functions, while allowing the receiver wavefield to propagate at steep (and overturning) angles to either side of the acquisition array where required. One consequence of using a 2D elliptic coordinate system is that the corresponding extrapolation wavenumber is specified by only a slowness model stretch. Thus, high-order implicit finite-difference (FD) extrapolators with accuracy up to 80 from the extrapolation axis Lee and Suh (1985) can be used to propagate wavefields, readily enabling accurate imaging of overturning waves at a cost competitive with Cartesian downward continuation.
This paper begins with a discussion as to why elliptic meshes are a natural coordinate system choice for shot-profile PSDM. We then develop an extrapolation wavenumber appropriate for propagating wavefields on 2D elliptic coordinate systems. We present a zero-offset example illustrating the ability of the scheme to image overturning wavefields. We then present prestack migration examples for the SMAART JV Pluto 1.5 and BP velocity benchmark data set and conclude with a brief discussion on the advantage of elliptic over more dynamic coordinate systems. Finally, Appendix A presents the wavenumbers for two 3D elliptic coordinates systems.