So far I have described plane wave synthesis along a flat datum
either at the surface or at a depth level.
In order to synthesize a plane-wave along an irregular
datum, we need an extrapolation algorithm that
extrapolates wavefield from an irregular datum.
Suppose that we want to have a
downgoing wavefield along an
irregular reflector *r*=*z*(*x*).
This wavefield can be obtained by propagating
a certain wavefield at the surface, ,to the irregular reflector *r*=*z*(*x*), as follows:

(9) |

(10) |

Therefore, to synthesize a plane-wave along an irregular reflector,
we need to formulate the wavefield extrapolation
from a flat surface to an irregular surface
as a forward operator, *W*(*r*,*z _{0}*),
and find its adjoint,.Those extrapolations are formulated
by modifying a standard depth extrapolation operator
and are explained in Appendix .

Another case that requires an extrapolation
similar to that described above is one in which
we synthesize a plane-wave at a depth level
using shot gathers collected along a nonflat surface.
In this case, we need to define the forward operator
as an extrapolation from a flat datum to an irregular datum,
*W*(*z*_{n},*r*), and as well as define its adjoint operator, .These operators are easily derived by a modification
of the algorithm shown above in this section.

2/5/2001