Introduction

Riemannian wavefield extrapolation (RWE) casts the problem of wavefield extrapolation in a framework independent of a particular coordinate system Sava and Fomel (2003). A practical implementation of RWE, though, requires specifying the computation domain on which to perform extrapolation. One judicious choice is a rayfield where the natural extrapolation direction is stepping in time along an individual ray. Rays in simple media are characterized by smooth curves, and are regularly distributed and (usually) triplication-free. However, rays in more complex media often exhibit tortuous behavior, are irregularly distributed, and are full of triplications. These characteristics, especially triplication, are problematic for the calculation of the derivatives necessary for RWE. Accordingly, a procedure must exist to minimize these adverse effects.

One solution to the ray-coordinate triplication problem is to use velocity models sufficiently smooth such that computed rayfields are triplication-free. The true velocity model is then mapped to the calculated smooth rayfield and RWE is performed as usual. Often, through, the amount of smoothing required to meet this objective causes rays to deviate significant distances from their true ray paths. This spatial relocation leads to a degradation of ray-coordinate extrapolation operator fidelity, because of the non-conformal orientations of the wavefront and extrapolation axes. However, operator accuracy may be partially restored by updating the ray-coordinate system to a basis better aligned with true extrapolation direction. One way to accomplish this is to extrapolate an initial wavefield on an initial ray-coordinate system, and to use this result to calculate a corresponding phase-rayfield Shragge and Biondi (2003). This rayfield then forms an improved coordinate basis on which to extrapolate wavefields using RWE.

This short note describes a procedure for updating ray-coordinate systems using phase-rays and RWE. This procedure differs from that of companion paper Shragge and Sava (2004), which presents a recursive bootstrap procedure for calculating a ray-coordinate system on-the-fly using the wavefield phase gradient of the previous few extrapolation steps. This short note is comprised of an overview of the processing flow required to generate the updated phase-rayfield, and examples of the method's application using the Sigsbee 2A velocity model.