Migration and data synthesis may be envisioned
in ( *z*' , *t*' )-space
on the following table, which contains the upcoming wave *U*:

(48) |

(49) | ||

(50) | ||

(51) |

The best-focused migration need not fall on the 45 line as depicted in (48); it might be on any line or curve as determined by the earth velocity. This curve forms the basis for velocity determination. You couldn't determine velocity this way in the frequency-domain.

The equation for upcoming waves *U* in
retarded coordinates (*t*' , *x*' , *z*' ) is

(52) |

(53) |

Now this partial-differential equation will be discretized
with respect to *t*' and *z*'.
Matrix notation will be used, but the notation does not refer to matrix algebra.
Instead the matrices refer to differencing stars that may be placed
on the -plane of (48).
Let denote convolution in -space.
A succession of derivatives is really a convolution, so
the concept of is expressed by

(54) |

(55) |

The sum of the two operators always has in the form

(56) |

Given the three values of *U* in the boxes, a missing one, *M*, may be
determined by either of the implied two operations

(57) |

(58) |

are unstable.
It is obvious that there would be a
zero-divide problem if *s* were equal to 0,
and it is not difficult to do the stability analysis
that shows that (58)
causes exponential growth of small disturbances.

It is a worthwhile
exercise to make the zero-dip assumption ( *k*_{x} = 0 ) and use
the numerical values in the operator of (56) to fill in the elements
of the table (48).
It will be found that the values of *u*_{t} move
laterally in *z* across the table with no change,
predicting, as the table should, that *c*_{t} = *u*_{t}.
Slow change in *z* suggests that we
have oversampled the *z*-axis.
In practice, effort is saved by sampling the *z*-axis
with fewer points than are used to sample the *t*-axis.

10/31/1997