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 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 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.