Depth extrapolation of the one-way wave equation in three dimensions can be
carried out using finite-difference methods in the domain.
The implicit formulation of the 3-D problem
produces a block tridiagonal system of
equations Claerbout (1985). Since the direct solution of this system
is an expensive process,
the usual technique used in carrying out the extrapolation is the splitting
technique. Splitting in three dimensions
is based on an approximation that enables
the full 3-D operator to be realized by applying the 2-D operator
along *x* for all *y* and then along *y* for all *x*.
This technique yields an operator that is anisotropic,
with the greatest error at the 45 degree azimuth.

In Yilmaz's book 1988, we found the reference to Ristow's four-pass technique Ristow (1980), in which the forty-five degree finite-difference operator was applied not only along the two conventional axes but also along the axes rotated to 45 degree azimuth from the original axes to compensate the greatest error at 45 degree azimuth. This paper tests this method and compares the results of the two-pass method and the four-pass method. In the future, we plan to compare it to the Hale-McClellan method.

11/16/1997