Theory of 3-D angle gathers in wave-equation seismic imaging

Algorithm II: Post-migration angle gathers

The second algorithm follows from equation (16). It applies after the imaging has completed and consists of the following steps applied at each common-image location:

1. Generate and store local offset gathers. In the double-square-root migration, the local offsets are immediately available. In the shot gather migration, local offsets are generated by cross-correlation of the source and receiver wavefields.
2. Estimate the dominant local structural dips at the common image point by using one of the available dip estimation methods: local slant stack, plane-wave destruction, etc.
3. After the imaging has completed, transform local-offset gathers into the slant-stack domain either by slant-stacking in the $\{z,h_x,h_y\}$ physical domain or by radial-trace construction in the $\{k_z,k_{h_x},k_{h_y}\}$ Fourier domain (Sava and Fomel, 2003).
4. Using estimated dips, convert slant stacks into angles by applying equation (16). The mapping from offset-depth slopes to angles is illustrated in Figure 4.

The last two steps can be combined into one. It is sufficient to compute the effective offset $\hat{h} = \sqrt{h_x^2+h_y^2+(h_x z_y - h_y z_x)^2}$ and apply the basic 2-D angle extraction algorithm to the effective offset gather.

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Figure 4. Mapping from the offset slope plane to angles according to Algorithm II. Zero slopes map to zero (normal-incidence) angle.

The second method is applicable to selected common-image gathers, which can be spread on a sparse grid. The local offset gathers need to be computed and stored at all depths. The method works independent of the velocity. The main disadvantage is the need to estimate local structural dips. In the common-azimuth approximation, only the cross-line dip is required (Biondi et al., 2003). In the 2-D case (zero cross-line dip), the method is dip-independent (Sava and Fomel, 2003).

Theory of 3-D angle gathers in wave-equation seismic imaging