Kirchhoff integral extrapolation and migration of nonplanar data

Abstract

The form of the Kirchhoff integral commonly used for migration of seismic data assumes that the observation surface is a flat, horizontal plane. This restriction is not necessary in theory or practice. A useful integral expression can be obtained for an arbitrary observation surface by use of the Kirchhoff approximation for relating the field on the observation surface to its normal derivative. Kirchhoff migration is usually carried out by an integral that represents summation over receivers for a given shot. Summation over shots is not explicitly handled. By invoking the principle of reciprocity and applying the Kirchhoff integral twice, one arrives at an integral expression that explicitly represents shot and receiver summation, contains the factors needed to compensate for nonplanar survey geometry, and can be used for wave extrapolation as well as migration. Adaptation of this integral to handling 2-D or line survey data is straightforward and, with the usual approximations concerning spherical wave spreading, it accomplishes 2-D wave extrapolation and migration. By tabulating both average and rms velocity functions, the Kirchhoff integral can be used to handle cases where the velocity changes vertically and the survey surface has vertical relief. The approximations involved are only those usually required in calculations of rms velocity traveltime. Migration is illustrated by treatment of two model data sets. One data set has shots and receivers extended horizontally with the receiver path undergoing strong elevation changes and the shot path flat. Migration of this data set illustrates the effect of the geometrical weighting factors and the effectiveness of the rms velocity approximation. The second data set is a vertical seismic profile (VSP) with shots extended horizontally and receivers extended vertically down a single well. Migration of this VSP data yields a useful image for a reasonable distance away from the well. Migration of nonplanar survey data is quite practical with geometrical artifacts that are manageable. The technique has use for surveys taken over nonplanar topography, for imaging from VSP data, and in a dual role for treatment of velocity anomalies, i.e., first as a wave extrapolation procedure to move below the anomaly and then as a migration procedure applied to the nonplanar extrapolated data.