Figure 3

We can apply gapped predictive deconvolution on this (*g*1',*g*2') in Figure 3a.
Conjugate mapping all the way from the
deconned Abelian group back to the space-time domain gives the
water-related reverberations attenuated section.

Two problems with this mapping are that it requires two more
(forward and conjugate via (*T*,1/*V*_{nmo}^{2}) ) mappings,
while the amplitude in the (*g*1',*g*2') domain is less regular
than in the second method described
below. So we move on to the second mapping method, a direct one.

The other way to go to (*g*1',*g*2') is to do velocity transform mapping directly
from the space-time domain. The conjugate mapping from (*g*1',*g*2') to material
parameters consists of:

- 1.
*T*=*g*1'- 2.
*V*_{nmo}^{2}=*g*2'/*g*1' +*V*_{water}^{2}.

Considering the velocity uncertainty, for a point
in (*g*1',*g*2') domain, we use also its left and right neighbors along the
*g*2' axis to invert for material parameters, yielding a *T*, and three
velocities squared.
We apply velocity transform by NMO Kirchhoff summation
over the hyperbola strip defined by the three velocities squared.
The weight given to each summing element is defined by a triangle weighting
function, an approximate to a Gaussian function (Jedlicka, 1989).
The apex of the triangle is at the NMO time of the median velocity squared.
The area under the triangle is ,i. e., we are honoring the 2-D Green's function.
The result of summation gives the value at
the corresponding point in (*g*1',*g*2') domain.
The conjugate mapping is the spraying out of the
point value in (*g*1',*g*2') domain into the NMO hyperbola strip
Kirchhoff trajectory in space-time domain (Claerbout, 1991).
One caution is that the coordinate *g*2'=0 must be sampled
because it directly samples the water bottom multiples.

The problem with this velocity transform mapping method
arises from the different
velocity sensitivity in the relation *V*_{nmo}^{2} = *g*2'/*g*1' + *V*_{water}^{2}
between shallow and deep parts. When *g*1'=*T* is small,
a small change in *g*2' results in a big change in *V*_{nmo}^{2}.
We discard the negative *V*_{nmo}^{2} as computed.
As a result, the velocity is coarsely sampled at the shallow part, whereas
the velocity is densely sampled at the deep part.
We apply a weighting function 1/*V*_{nmo}^{2} to balance between the
number of small and
large velocity values used, and to underweight the ridiculously high
velocities as computed at the shallow part.
And because the 2-D Green's function favors the near-offset strongly, after
conjugate inverse velocity transform, the far offset data is attenuated.
We include an extra weighting function ,
where is the
Kirchhoff obliquity factor.
From our experiences, the results
are acceptable when *T*>0.3 *second*.

11/18/1997