Prestack Equivalent Offset Migration: The Basis

A method of prestack time migration is presented that is simpler, faster, and provides better velocity<br>information than conventional Kirchhoff methods. It is based on prestack Kirchhoff time migration and can be<br>applied to both 2-D and 3-D data. The method is divided into two steps, a gathering process that forms common<br>scatterpoint (CSP) gathers, followed by an imaging process using a Kirchhoff moveout correction that is<br>performed independently on each CSP gather. The CSP gathering process sums input traces into equivalent<br>offset bins in each CSP gather with no time shifting. The equivalent offset is defined by an exact hyperbolic<br>simplification of the double square root (DSR) equation of prestack time migration. A CSP gather is similar to a<br>CMP gather as both contain offset traces, and both represent a vertical array of scatterpoints. CSP gathers can<br>be formed at any arbitrary location, have high fold in their offset bins, and have a much larger offset range due<br>to the gathering of all input traces within the migration aperture.<br>After the CSP gathers have been formed, conventional velocity analysis estimates accurate prestack<br>migration velocities. The high fold and large offset of the CSP gathers provide better focusing for improved<br>velocity analysis. The imaging process collapses each CSP gather into a single migrated output trace. It is<br>performed as a Kirchhoff process, which consists of scaling, filtering, normal moveout (NMO) correction, and<br>stacking. Significant computational savings result from delaying arithmetic operations on the input samples<br>until after a CSP gather has been formed. This space-time domain method is suitable for uneven geometries,<br>may be adapted to migrate from topography, enables velocity analysis at random locations, and permits<br>prestack migration of a 3-D volume into an arbitrary 2-D line. An accompanying paper describes how the<br>equivalent offset method is extended to rugged topography, prestack migration of converted-wave (P-S) data,<br>compute residual statics before NMO correction, perform anisotropic prestack depth migration, and prestack<br>migrate vertical receiver array data from VSP’s or vertical marine cables.