There are (at least) two possible ways to circumvent this issue. The first procedure involves using precomputed wavefields to train the phase-ray coordinate system by: i) establishing the longer wavelength rayfield structure by raytracing in a background velocity model using a broadband solver; ii) calculating an initial monochromatic wavefield solution using the background rayfield as the coordinate system; iii) computing an updated ray-coordinate system from the previous wavefield solution; and iv) calculating an updated wavefield on the improved phase-ray coordinate system. This procedure is addressed in companion paper Shragge and Biondi (2004) in this report.
A second approach is to use wavefields parameterized in phase-ray coordinates, rather than Cartesian, to dictate the direction of the next rayfront step. For judiciously chosen steps, both the rayfield direction and resulting wavefield evolves slowly, and ray directions differ by only small, incremental amounts in a neighborhood of . Hence, the orientation of previous few ray steps provide a good estimate of the required ray direction at the present step. Thus, a phase-ray coordinate system may be computed on the fly using the magnitude of the wavefield phase gradient of the previous few steps and the local velocity function.
Using a wavefield parameterized in ray-coordinates to generate the underlying coordinate system requires that the governing phase-ray equations are transformed accordingly. Fortunately, the magnitude of a scalar field gradient remains invariant to coordinate transformation, and is related through a change of variables,
Computing a weighted average direction from the previous M steps requires saving M+1 previous wavefield steps. The discrete version of equation (12) for ray step at index t, , is,
The use of previous wavefield solutions to compute solutions to equations (12) naturally gives rise to a broadened finite difference stencil. Figure 2 presents the finite difference stencil for the M=2 case that has second-order accuracy in and .
Figure 2 Finite difference stencil for calculating wavefield solution at present step using the previous two steps (M=2). The solid square represents the location of the desired rayfield solution, and the in-filled circles connected by lines are the points contributing to the solution point.