Let's examine a more complicated media. In any situation there is an optimal value
*v*_{opt} of stacking velocity that supplies the order of touching .Let us put

(141) |

One can easily check that condition (141) supplies the order of touching
*r*=2. But both stacking hyperbola or common-mid-point travel-time curve are even
functions (with respect to *h*) so third derivatives of both functions at *h*=0
are equal to . It means that *r*=3 and

We shall give some examples of optimal stacking velocities:

- if
*v*(*x*,*z*) is constant and reflector is curved, then where is a slope of the reflector at the base of the normal ray from the point*X*. - If
*v*(*x*,*z*)=*v*(*z*) and , then (H is the reflector's depth). - If
*v*(*x*,*z*)=*v*(*t*) and a reflector is curved, then where .

In practice, the optimal value of extreme velocity can be achieved as a result of velocity spectra analysis.

So, we have

Accurate optimization of migration velocity for prestack migration is also
possible. Let us consider the example of a flat horizontal reflector in vertically
inhomogeneous velocity *v*=*v*(*z*). It can be shown that at *v*_{m}=*v*_{rms} the second
derivative of the function *z*=*h*(*x*) which describes the shape of the reflector's
image equals to (under *x*=*s*). The function *h*(*x*) is symmetrical with respect
to the point *x*=*s* and it means that again *r*=3 and .

It is interesting to note that the operator gives the good order but wrong depth ( is an average
velocity). On the other hand, the operator gives the order but the correct depth *H ^{'}*=

To perform post-stack migration for this model we can use the operator

which simultaneously gives
and *H ^{'}*=

When there is no symmetry in the medium, optimal migration velocity
in prestack migration guarantees only *r*=2. So in general,

1/13/1998