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Bootstrapping the chicken to get the egg

Having parameterized the phase-ray equations in ray-coordinates, and specified a method for updating rayfront directions, it is possible to detail the bootstrap method that forms the core of adaptive phase-ray extrapolation procedure. Figure 3 presents a flowchart representation of the bootstrap procedure.

 
flow
Figure 3
Flow chart of the bootstrap procedure
flow
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Rayfields must be computed prior to wavefield extrapolation. Accordingly, the coordinate system is first initialized by assuming the first M+1 steps using an educated guess of where wavefront energy will propagate. Two examples are an expanding circular mesh for a point source (illustrated in Figure 1) or a tilted coordinate system for a dipping plane-wave source (illustrated in Figure 4).

 
dipstart
Figure 4
The first 4 steps of an initial coordinate mesh appropriate for initializing a dipping plane-wave source. At coordinate locations above the ground surface the velocity model is assumed to be constant so that extrapolated energy enters the model as a monochromatic plane-wave (i.e. in both $\omega$ and kx).
dipstart
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After coordinate system initialization, M wavefield extrapolation steps are carried out to generate the required M+1 step wavefield. The bootstrap process is a loop around three separate calculations: i) ray step $\Delta r_i$from the previous M wavefield steps; ii) rayfield Jacobian spreading, J, and associated functions; and iii) wavefield ${\cal U}$ at the current step. The final step involves interpolating the wavefield from the ray to the Cartesian coordinate basis, and is done independently after extrapolation.


next up previous print clean
Next: Phase-ray extrapolation examples Up: Theory Previous: The chicken and the
Stanford Exploration Project
5/23/2004