Radiation Patterns and Free-surface

Two additional factors that warrant comment are the effects of source radiation patterns and the free-surface (see sketch in figure 4). A dipolar coordinate system can partially account for these effects. I generate the initial source wavefield by extracting the direct arrival wavelet from a receiver nearby the source point and spreading it across the $180^{\circ}$ arc to form a radially symmetric wavefield. To impart a more realistic radiation pattern, I then introduce an angular cosine radiation amplitude filter that helps ensure that the wavefield has zero amplitude (at least initially) at the free surface.

 

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Figure 4 Sketch showing factors complicating modeling wavefield propagation with one-way extrapolation operators. Properly modeling wavefields requires generating multiple arrivals - direct, wide-angle reflections and ghosts - as well as handling the free-surface and source radiation patterns.

Free-surface effects can be approximately incorporated by allowing waves to propagate in a vertically mirrored velocity model. Figure 5 illustrates the expected arrivals from the mirroring procedure. The top panel shows the expected reflected and ghost arrival polarities, while the bottom shows the rays modeled with the mirrored velocity approach. Note that negating the radiation pattern in the mirrored velocity panel generates an effective free-surface ghost, including the R=-1 free-surface reflection coefficient. First-order multiples are also present, but have incorrect polarities that could cause problems in the inversion. Multiples arising from deeper reflectors, though, arrive later and are routinely windowed out from the data.

 

mirror Figure 5 Sketch showing the velocity mirroring process that enables computation of the free-surface reflected ghost. Top panel: Expected ray arrivals. Bottom panel: Calculated arrivals using the mirrored velocity approach.