** Next:** Stability of the differential
** Up:** The craft of wavefield
** Previous:** Fractional integration and constant

Stable extrapolation can be assured by preserving certain symmetries.
It will be shown that stability is
assured in both the differential equation

| |
(80) |

and its Crank-Nicolson approximation
| |
(81) |

provided that is
a positive definite (actually, semidefinite) matrix.
When stability was studied in the previous section
the operator *R* was a scalar *Z*-transform.
Because *Z*-transforms were used,
the mathematics of that section was particularly suitable
for time domain migrations.
Because *R* was a scalar,
the mathematics of that section was particularly suitable
when data has been Fourier transformed over *x*.
Here we will focus on the
*matrix*
character of *R*.
Thus we are concerning ourselves with migration in the *x*-domain.
Our purpose in doing this theoretical work
is to gain the ability to write a ``bulletproof'' program
for migrating seismic data in the presence of lateral velocity variation.
As an example,
the familiar 45 extrapolation equation
will be put in the bulletproof form.
This section, combined with the previous one,
gives a general theory for stable migration in (*t*,*x*)-space.

** Next:** Stability of the differential
** Up:** The craft of wavefield
** Previous:** Fractional integration and constant
Stanford Exploration Project

10/31/1997