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There is a notable drawback from the approach
described above.
The operator can be quite
costly, We are
doing nx*ny AMO transforms
for every output (hx,hy). If
we are only interested in a
common azimuth dataset , the cost
is acceptable as long as *ny* is
fairly small.
If we want any cross
line offset output
offset the cost isn't acceptable.
In addition is a modified version (because
of the AMO transform) of a small 2-D box
car filter. If you desire additional smoothness
in the in-line offset direction (to
suppress amplitude variations) we must try
a different approach.
Biondi and Vlad (2001) proposed
reducing the dimensionality of the problem
by ignoring the azimuth direction. They
added a smoothness constraint to the
problem by applying a Leaky derivative operator between AMO transformed (t,cmpx,cmpy)
cubes setting up the minimization,

| |
(7) |

where controls the weighting between the two goals.
They preconditioned the problem with the inverse of ,leaky integration between AMO transformed cubes and
applied the same Hessian approximation to obtain the
approximation
| |
(8) |

where is the inverse of and
| |
(9) |

Clapp (2003b) noted
that ignoring the azimuth removed some of the advantage of
using the AMO operator and suggested that
should be applying polynomial division
with a 2-D filter operating in the the (hx,hy) plane.
For this exercise I chose a small helical derivative
for my 2-D filter Claerbout (1999).