Delayed-shot migration in TEC coordinates

Discussion

Relative computational cost is one important metric to consider when comparing the migration algorithms in different coordinate systems. In the above tests, padding in the crossline direction tended to be the most important factor in determining the migration run time. One benefit of the TEC geometry is its naturally outward-expanding mesh in the elliptical direction that effectively increasing the migration aperture. Thus, TEC migrations usually require less zero-padding in the crossline direction relative to CC geometries. I performed the TEC migrations on meshes with inline-by-crossline-by-depth grids of 720x324x400. Migrations in CC geometries required a 720x512x400 mesh in order to achieve similar crossline aperture, which resulted in a fairly significant additional computational overhead.

Table 3 shows the comparative costs for various TEC and CC migration runs for both the shot-profile and delayed-shot migration styles. I used 72 data points in specifying each median runtime times for the four different migration runs. The test migrations indicate that the TEC geometry migrations were faster than the those in Cartesian tests (for equivalent effective aperture), with 29% and 21% computational cost reduction for the shot-profile and inline delayed-shot migration strategies, respectively.

Table 3: Run-time comparisons for the delayed-shot migration and shot-profile styles in the tilted elliptical-cylindrical and Cartesian coordinate systems.

Migration	Coordinate	Mean run
style	system	time (hrs)
Plane-wave	Tilted elliptical cylindrical	37.2
Plane-wave	Cartesian	45.0
Shot-profile	Tilted elliptical cylindrical	15.5
Shot-profile	Cartesian	20.0

One question worth addressing is how far can the TEC sampling be reduced before imaging artifacts become apparent? As one moves outward between successive extrapolation surfaces, the TEC geometry expands at increasingly larger step sizes. Fortunately, most realistic velocity models have velocity increasing with depth, causing the wavelengths of the propagated waves to lengthen. This phenomenon acts as a natural wavefield filter that, in most cases, prevents wavenumbers from aliasing (except near-surface in the grid extremities). A good rule-of-thumb is that one must ensure that the grid point of TEC coordinate system mesh does not go below one grid point for every two CC grid points in each direction; however, maintaining this relationship throughout the image volume is not a straightforward task. Additional work on the craft of 3D coordinate-system interpolation is necessary and would likely help restore some of the absent high frequency information.

An additional consideration of parameter choice is the interpolation window over which the surface wavefields are injected onto the TEC coordinate mesh. Not using a sinc-based interpolation over the near-surface depth axis can lead to significant artifacts; however, choosing too large of a window will blend information from different extrapolation steps leading to smoother and lower frequency images. Figures 7-10 show the result of a somewhat overcautious parameter choice (interpolating wavefields three additional depth steps) that led to the lower spatial wavenumber content of the TEC images relative to the CC images. I assert that his effective low-pass filtering can be reduced by interpolating only one or two additional steps in depth.

Delayed-shot migration in TEC coordinates