Near-field vs. far-field Kirchhoff datuming

Figure  is a comparison of two Kirchhoff upward continuations. In Figure a the near-field term is retained, and in Figure b the far-field approximation is made so that equation () is applied directly. Both results are kinematicly equivalent, but there is a small difference in amplitude and phase behavior, especially for small time shifts $\tau$ near the center of the plots where the topography is close to the output datum. The result of migrating Figures a and b is presented in Figure c and d. There is not much discernible difference between these two plots.

Figure  shows that as the data are upward continued to a datum that is 100 m above the topography, the amplitude and phase behavior of the near-field and far-field Kirchhoff extrapolations converge. This is further illustrated in Figures  and  where traces from two lateral locations are compared after upward continuation with the near-field and far-field Kirchhoff datuming operators. The top traces in Figure , where the time shift $\tau$is small, exhibit a substantial difference in amplitude and phase. With increasing extrapolation distance, the differences are not as pronounced. In Figure  I compare two traces at a location where the synthetic wavefield is simpler and where the extrapolation distance is longer because the recording surface is far from the output datum.

From these examples, I conclude that for most cases, the far-field approximation to the Kirchhoff integral is adequate. This is especially true if the data are subject to wave-based processing (in this case, time migration) after datuming. This is significant because it is computationally more efficient to implement Kirchhoff datuming if the near-field term is dropped.

 

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Figure 12 Comparison of (a) near-field Kirchhoff datuming and (b) far-field Kirchhoff datuming. Comparison of migrated (c) near-field Kirchhoff datuming and (d) far-field Kirchhoff datuming. The data have been upward continued to a datum just above the highest topography.

 

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Figure 13 Comparison of (a) near-field Kirchhoff datuming and (b) far-field Kirchhoff datuming. The migrations of the top two panels are presented in c and d. The data have been upward continued to a datum 100 m above the highest topography.

 

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Figure 14 Comparison of traces at horizontal position of 0.5 km for near-field and far-field Kirchhoff datuming. Near-field result is plotted as a continuous line, far-field result is plotted as a dashed line. The plots are for output datum levels of 210 m (top), 300 m (middle), and 400 m (bottom).

 

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Figure 15 Comparison of traces at horizontal position of 0.25 km for near-field and far-field Kirchhoff datuming. Near-field result is plotted as a continuous line, far-field result is plotted as a dashed line. The plots are for output datum levels of 210 m (top), 300 m (middle), and 400 m (bottom).