Prestack Kirchhoff inversion of common‐offset data

Abstract

In trying to resolve complex geologic structures, the pitfalls in employing the CDP method become evident. Additionally, stacking multioffset traces corrupts the amplitudes necessary for stratigraphic analysis. In order to preserve whatever structural and amplitude information is in the data, prestack processing should be performed. Given common‐offset data and the velocity above a reflector, prestack acoustic Kirchhoff inversion resolves the location of the interface. When amplitude information has been preserved in the data, the method additionally calculates the reflection coefficient at each interface point. For band‐limited seismic data, the inversion operator produces a sinc‐like picture of the reflector, with the peak amplitude of this band‐limited singular function equal to the angularly dependent reflection coefficient. The inversion development is based upon high‐frequency Kirchhoff data which are inserted into a general 3-D inversion operator. Asymptotically evaluating the four resulting integrals by the method of four‐dimensional stationary phase permits an inversion amplitude function to be chosen so that the inversion operator produces a singular function of support on the reflector, weighted by the reflection coefficient. Specializing the three‐dimensional inversion operator to two and one‐half dimensions allows for processing of single lines of common‐offset data. Synthetic examples illustrate the accuracy of the method for constant‐velocity Kirchhoff data, as well as the problems in applying constant‐velocity data to multivelocity models.