The split-step method is a mixed domain operator (-x) that adapts a lateral velocity variation. Different approximations for split-step migration have been published (Biondi and Palacharla (1996); Herbert (1992); Huang and Fechler (1997); Kessinger (1992)). Stoffa et al. (1990) introduced the split-step method to migrate post-stack data using one reference velocity. In contrast, the extended split-step algorithm uses more than one reference velocity to accommodate lateral velocity variations, similar to the way that phase-shift-plus-interpolation works Gazdag and Sguazzero (1984).

The split-step algorithm solves the wave equation in -space coordinates. The downward continuation of a wavefield using a split-step algorithm has two parts: the wavefield is first downward continued with a phase shift defined by the DSR operator with a constant reference velocity (equation 2, or focusing term); this is then followed by a vertical time shift correction in the space domain applied to the continued wavefield proportional to the contrast in slowness.

In a strong lateral velocity gradient, the split-step migration does not properly image
steep reflectors when *k*_{z} wavenumbers are
limited to avoid evanescent energy (i.e., solving the one-way wave equation). Two consecutive reference
velocities define the interval where the required downward continued wavefield for
the mapping velocity needs to be interpolated. In general, these wavefields
are downward continued by applying a phase shift defined by the vertical
depth-wavenumber, *k*_{z}, as a real function of the migration dips. This last
assumption limits the interpolation of the continued wavefields for a
velocity between two reference velocities because dips at low velocities
correspond to dips at higher reference velocity, and those dips could be lying in the evanescent region of the higher reference velocity Kessinger (1992).

In this paper, we show that including the evanescent energy in the *k*_{z}-domain
helps to obtain an accurate interpolated downward continued wavefield for
every migration depth step. Three different data sets are used to make
comparisons between extended split-step migration with and without
evanescent energy.

In addition, we have an economical 3-D extended split-step prestack migration based on the common-azimuth downward continuation of the wavefield. Common-azimuth migration is based on the stationary-phase approximation of the 3-D DSR in every downward continuation of the wavefield, resulting in a DSR operator independent of the in-line-offset wavenumber (Biondi and Palacharla, 1996).

7/5/1998