Recursive Kirchhoff continuation

Kirchhoff datuming can be applied recursively, as in the case of finite-difference and phase-shift datuming. This recursive formulation is embodied in equation (), where the W_i operators represent the Kirchhoff extrapolation operator between small depth intervals $\Delta z$. Since the depth intervals are small, the time delay $\tau$ is short and the computation is definitely in the near field, so it is important to retain the near-field term. Because the extrapolation distance is short, fewer traces can be taken into the integral operator. However, the short extrapolation distance makes the Kirchhoff summation trajectory very steep, necessitating a high degree of operator anti-aliasing for data right at the apex of the summation trajectories. When the depth step is small, the Kirchhoff operator approaches the limit of becoming an explicit finite-difference extrapolator, and all of the difficulties of designing such an operator begin to manifest themselves.

The result of upward continuing the synthetic data with the recursive Kirchhoff datuming algorithm is shown in Figure . Ten traces were used for every application of the Kirchhoff extrapolator and the same number of depth levels (21) were used as in the phase-shift and finite-difference results. The datuming and migration results are much more low frequency because of the high degree of anti-aliasing required, and the steep dip resolution has decreased because of the limited width of the extrapolator. The migrated image in Figure b looks more like the image after finite-difference datuming (Figure c) than after Kirchhoff datuming (Figure b).

The importance of keeping the near-field term is illustrated in Figure . When the far-field approximation is made, the recursive extrapolation is overcome by strong artifacts.

 

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Figure 16 (a) Recursive Kirchhoff datuming and (b) migration of the recursively redatumed synthetic.

 

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Figure 17 Illustration of the importance of including the near-field term in recursive Kirchhoff extrapolation. (a) When the near-field term is included, the extrapolation is successful, (b) when the far-field approximation is made, the recursive extrapolation is dominated by artifacts.

The recursive Kirchhoff extrapolator results in poorer results than any of the aformentioned methods and is very inefficient computationally. I have presented it merely to demonstrate how the datuming methods can be linked. In practice there are much more efficient methods of recursive explicit extrapolation.