Depth imaging using slowness-averaged Kirchhoff extrapolators
Hugh D. Geiger, Gary F. Margrave, Kun Liu, P. F. Daley
Recursive Kirchhoff wavefield extrapolation in the space-frequency domain can be thought of as a simple convolutional filter that calculates a single output point at depth z+dz using a weighted summation of all input points within the extrapolator aperture at depth z. The desired velocity values for the extrapolator are the ones that provide the best approximation of the true phase (propagation time) of the seismic wavefield between the input points and the output point. Recursive Kirchhoff extrapolators can be designed to handle lateral variations in velocity in a number of ways: a PSPI-type extrapolator uses only the velocity at the output point, a NSPS-type extrapolator uses the velocities at the input points; a SNPS-type extrapolator incorporates two extrapolation steps of dz/2 where the first step uses the velocities at the input points (NSPS-type) and the second step uses the velocity at the output point (PSPI-type); while the Weyl-type extrapolator uses an average of the velocities between each input point and the output point. Here, we introduce the PAVG-type extrapolator, which uses velocity values calculated by an average of slowness along straight raypaths between each input point and the output point. A simple synthetic with a lateral step in velocity shows that the PAVG Kirchhoff extrapolator is very close to the exact desired response. Tests using the Marmousi synthetic data set suggest that the extrapolator behaviour is only one of many considerations that must be addressed for accurate depth imaging. Other important considerations include preprocessing, aperture size, taper width, and imaging condition.