Wavefield extrapolation in phase-ray coordinates

Jeff Shragge and Biondo Biondi

jeff@sep.stanford.edu, biondo@sep.stanford.edu

ABSTRACT

A ray theoretic formulation is developed that allows rays to be traced directly from existing solutions to the Helmholtz equation. These rays, termed phase-rays, are defined by the direction normal to surfaces of constant wavefield phase. Phase-rays exhibit a number of attractive characteristics, including triplication-free ray-fields, an ability to shoot rays forward or backward, and an ability to shoot infill rays for ensuring adequate ray density. Because of these traits, we use phase-rays as a coordinate basis on which to extrapolate wavefields using the generalized coordinate system approach. Examples of wavefields successfully extrapolated in phase-ray coordinates are presented, and the merits and drawbacks of this approach, relative to conventionally traced ray coordinates, are discussed.

Ray theory is routinely applied to generate characteristics to solutions of the Helmholtz equation. The usual ray theoretic approach introduces an ansatz representation of the wavefield solution into the Helmholtz equation to yield coupled eikonal and transport equations. Computation of eikonal equation solutions is usually facilitated by the introduction of a high-frequency approximation that removes the amplitude dependence from the full eikonal equation. Consequently, conventionally computed rays are independent of frequency and often triplicate due to their broad-band nature.

In contrast, whenever independent solutions to the Helmholtz equation exist, full ray theoretic formulae may be used to trace rays Foreman (1989). These rays are termed phase-rays herein owing to a ray direction that is always orthogonal to surfaces of constant phase. One of their beneficial traits is that, unlike conventionally traced rays, computed ray-fields are caustic-free. A second advantage is that ray position is the only required initial condition for ray-tracing. However, one obstacle seems to restrict the use of the phase-ray formulation in seismic imaging problems: wavefield solutions to the Helmholtz equation must be known in advance of ray-field computation.

One situation where phase-rays (and conventional rays) are of use is in wavefield extrapolation in generalized (i.e. non-Cartesian) coordinates Sava and Fomel (2003). A natural set of coordinates for wavefield extrapolation is the general ray family represented by a ray direction and shooting angle(s). Wavefield extrapolation in ray coordinates thus uses a ray-field as the coordinate system on which to extrapolate wavefields. This approach transforms the physics of one-way wave propagation from the usual Cartesian grid so that it is valid in a ray coordinate system. Wavefield extrapolation operators are then applied more accurately because extrapolation can occur at lower angles to the ray direction than usually occurs with Cartesian coordinates. The result is then mapped from ray coordinates to Cartesian. Although ray domain extrapolated wavefields are generally more accurate, issues still remain when using conventionally-traced rays; in particular, how to robustly deal with infinite amplitudes at locations of coordinate system triplication.

Motivated by this issue, this paper examines the use of phase-rays as a coordinate system for wavefield extrapolation. The main advantage of phase-ray coordinates is that they are caustic-free, and thereby avoid complications arising from triplicating coordinates. The paper begins with a general discussion of ray theory and an approach for calculating phase-rays from solutions to the Helmholtz equation. Phase-ray examples from a salt body model are then presented, and are followed by results illustrating wavefield extrapolation in phase-ray coordinates for a Gaussian velocity perturbation model. Finally, a method is proposed for propagating broadband wavefields using frequency-dependent phase-ray coordinates.