Different algorithms have been proposed to perform DMO assuming a constant velocity medium. Yilmaz and Claerbout (1980) proposed a processing sequence called ``Prestack Partial Migration'' (PSPM) which represents the difference between the migration before stack and the conventional data processing scheme of NMO correction, stack and zero-offset migration.
Deregowski and Rocca (1981) proposed an integral formulation for DMO by considering the impulse response of the NMO+DMO operator as a transformation from constant-offset to zero-offset following raytracing arguments. This opened the road toward the development of the most flexible DMO method: DMO as an integral method. The kinematics of the impulse response are correctly determined by this method though the amplitude of the operator continues to be an area of research where different schools of geophysics do not reach an agreement.
Hale (1983) proposed a method to perform DMO via Fourier transforms which is kinematically accurate for all dips and offsets in constant velocity media. He also derived extensions to DMO processing to approximate slow velocity variations with depth. Hale's DMO method became the standard to which most DMO publications compare their. Zhang (1988) modified the Jacobian of Hale's DMO to obtain a true amplitude DMO operator.
Jakubowitz (1990) proposed an efficient version of DMO via Fourier transform by decomposing the data into a discrete number of dips, each processed separately with a dip-dependent stacking velocity. These dips are then added together after dip filtering so that each dip contributes only once to the final CMP stack. He also proposed in principle an extension to his algorithm for post-stack DMO.
Biondi and Ronen (1987) proposed an algorithm to perform DMO directly in shot profiles by multiplying the data in the Fourier domain by a time and space invariant shot-DMO operator.
These conventional DMO operators are kinematically correct when velocity is constant and they are approximately correct when velocity increases linearly with depth (Hale, 1983), but they cannot be applied when velocity varies laterally or when velocity varies relatively fast with depth and depth prestack migration is required to image the structure. In this case the migration to zero-offset (MZO) process cannot be split into NMO followed by DMO, but must be applied as a single process.