Wavefield resampling during Kirchhoff extrapolation

Gary F. Margrave and Hugh D. Geiger


Recursive Kirchhoff wavefield extrapolators can be used to both downward-continue a digital seismic wavefield and to interpolate it to new spatial locations. We present a brief review of the Kirchhoff theory and then three numerical experiments, in 2D, intended to demonstrate the capabilities of the Kirchhoff extrapolators. In the first two experiments the input wavefield is a 256 trace synthetic wavefield computed by upward extrapolating two impulses in a laterally variable velocity using the exact wavefield extrapolation theory. In a first test, the input wavefield is decimated to 128 regularly spaced traces and a reconstruction is attempted of the initial wavefield while downward extrapolating the decimated wavefield. This is compared to a downward extrapolation of the full 256 trace wavefield. The reconstructed wavefield is very similar to the full wavefield except for a slight increase in noise. The spectrum of the reconstructed wavefield has been extended beyond the spatial Nyquist for 128 traces; however, energy that was aliased in the decimation was not unaliased. This leads to the increased noise. A second test is similar except that the original 256 trace wavefield was downsampled to 162 traces chosen at random. Again a good reconstruction is obtained characterized by tight focusing plus noise. However, the noise is not easily attributable to aliased energy this time. Finally, a real shot record is used with results that are consistent with the conclusions obtained form synthetics. The recursive Kirchhoff approach is postulated to be especially useful for extrapolation of data acquired at irregular locations.

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