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

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).

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.