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Two additional factors that warrant comment are the effects of source radiation patterns and the freesurface (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 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.
Shotconfig
Figure 4 Sketch showing factors complicating modeling wavefield propagation with oneway extrapolation operators. Properly modeling wavefields requires generating multiple arrivals  direct, wideangle reflections and ghosts  as well as handling the freesurface and source radiation patterns.
Freesurface 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 freesurface ghost, including the R=1 freesurface reflection coefficient. Firstorder 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 freesurface reflected ghost. Top panel: Expected ray arrivals. Bottom panel: Calculated arrivals using the mirrored velocity approach.

 
Next: Forward modeling examples
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Stanford Exploration Project
1/16/2007