Anti-aliasing implications

Figure shows the impact on the image space of subsampling the shot axis by a factor of ten while migrating the flat reflector synthetic data described above. The left panel imaged with only every tenth shot, while right panel migrated shots at every receiver location. The shot-axis, which could be drawn at a 45^o angle up and to the right, shows inappropriate replications Rickett and Sava (2002). The data are modeled with sufficient receiver density, that this level of decimation does not alias the receiver gathers. This is corroborated by the absence of aliased energy in the upper-left and lower right quadrants.

 

alias-flat Figure 7 Left panel shows the wavenumber energy for a migrated flat reflector when using sources at every tenth receiver location. On the left, and all source locations.

From this simple example, we can see that the anti-aliasing restriction required for the decimated migration are sloped lines to remove energy from the upper-right and lower-left quadrants. In general, any of the four corners may experience aliased replications depending on the inequality between receiver and source sampling during acquisition. The form of the imaging condition in the Fourier domain as shown in equation 7 provides important insight into how to implement anti-aliasing criteria for shot-profile migration.

Limiting the image by neither constant k_x nor k_h will appropriately remove the aliased energy of Figure . Instead, one should limit the maximum bandwidth of both the $\hat{U}$ and $\hat{D}^*$wavefields. Table 1 provides a convenient display of this fact. This will maintain the center diamond of appropriate energy. If the anti-aliasing bandlimit is applied to the image space instead of the two wavefields used to calculate it, there are two important conclusions: 1) the bandlimit should be the same for both the offset and location axes, and 2) the limit is a diamond shaped, not circular, filter on the k_x-k_h plane.