## The theoretical basis for prestack migration by equivalent offset

### Gary F. Margrave, John C. Bancroft, Hugh D. Geiger

The method of prestack migration by equivalent offset (EOM) forms common scatterpoint (CSP) gathers for each migrated trace and then images those gathers with a migration algorithm. The major benefits are that the CSP gathers are formed by trace mappings at constant time and that trace binning can be conveniently done as the gathers are formed. Furthermore, the CSP gathers are very sensitive velocity analysis instruments. To provide a foundation in scalar wave theory, the Fourier dual algorithm to EOM, called equivalent wavenumber migration or EWM, is derived from Fourier migration theory. Both EWM and EOM are based on the algebraic combination of a double square root equation into a single square root. This result defines equivalent wavenumber or offset. EWM is found to be an exact reformulation of prestack f-k migration. The CSP gathers are shown to be formed by a Fourier mapping, at constant frequency, of the unmigrated spectrum followed by an inverse Fourier transform. The imaging expression (for each CSP gather) which results from this analysis is formally identical to post stack migration with the result retained only at zero equivalent offset. Through a numerical simulation, the impulse responses of EOM and EWM are shown to be kinematically identical. Amplitude scale factors, which are exact in the constant velocity EWM theory, are implemented approximately in variable velocity EOM.