The downward extrapolation of a zero-offset (stacked) time section by use of the finite difference method in the 2D Cartesian space coordinates and frequency domain, plus the concept of the exploding reflector model, gives a depth image of the earth's reflectivity interior, because the time events are migrated to their subsurface location. This processing is designed to image time patterns when there are velocity variations in both depth and lateral direction.
The one-way wave equation is extracted from the scalar wave equation first introduced by Claerbout in 1972. We derive extrapolation operators by performing a matrix finite periodic fraction expansion of the "square root operator". This square root appears naturally in the one-way wave equation.
The extrapolation operators that handle severe velocity variations in both depth and lateral directions are split in 45-degree operator. The stability analysis, made with absorbing side boundary conditions, gives a condition sufficient to guarantee the absolute stability of the extrapolation scheme.
Reflectors dipping steeply up to 55-degrees are properly migrated by the extrapolation operator generated by the first order of the finite periodic expansion, the so-called 45-degree approximation of the square root.
For complex structures, the conventional processing s abandoned for the prestack migration, processing which requires wide-angle approximations of the square root. The second order of the periodic fraction expansion, called the 65-degree approximation of the square root, is then used to downward extrapolate the common shot gathers.
STANFORD