To implement prestack phase-shift migration efficiently in *v*(*z*) media, we need to
construct a table of using ray tracing, where is the vertical two-way traveltime and
*t* is the two-way zero-offset traveltime.
Due to the lateral homogeneity assumption,
this table is constructed once and is applicable everywhere. The stationary point is evaluated by finding

(17) |

(18) |

To find the maximum of equation (18), I use a method described by Buchanan and Turner (1992).
This is a robust algorithm used to find a maximum or minimum of a function numerically
without the need to resort to evaluating
derivatives (similar in principal to the bisection method used to evaluate roots of functions).
Other methods, based on the finding the roots of the derivative of equation (18)
with respect to *p*_{h}, will most likely fail due to the sensitivity of these derivatives to typical
numerical errors associated with ray tracing.
Another table, , consisting of *T* for the solutions of equation (17) for given
*p*_{x}, time, and offset is, therefore, constructed. Again, such a table is applicable
everywhere in the medium due to the lateral homogeneity assumption considered here.
The cost of a prestack migration of any offset is about 10higher than that for a zero-offset algorithm, where *p*_{h}=0 is known,
and thus we do not need to evaluate it.
This relatively small additional cost reduces even further when large data sets are migrated and the
cost of precomputing the traveltime tables become insignificant.

11/11/1997