- 1.
*g*1=*T*(two-way, zero-offset travel time)- 2.
*g*2=*T*.*V*_{nmo}^{2}(Dix's parameter)- 3.
*g*3=*T*.*V*_{nmo}^{4}.*m*, which includes the anelliptic factor (*m*=1+4.*q*_{w}-*q*_{w}^{2}) in NMO equation- 4.
- log2 Hilbert envelope amplitude (in units of 0.25?)
- 5.
- Frequency-dependent attenuation.

At this stage, we implement only the first two attributes,
(*g*1=*T*, *g*2=*T*.*V*_{nmo}^{2}).
The addability
of *T* and *T*.*V*_{nmo}^{2} means that waterbottom multiples and peglegs lie
evenly sampled along each of these two axes. Then they lie evenly sampled
along parallel lines in the (*g*1,*g*2) plane, the slope being the water velocity
squared.
Figure 1 is a Solid elastic modeling dataset that has geology similar
to that of the Geco Barents Sea dataset we will study later.

Figure 1

We apply velocity transform by NMO Kirchhoff summation on this dataset
into the zero-offset two way travel time and
slowness squared domain (*T*,1/*V*_{nmo}^{2}) (Claerbout, 1992).
The zero-offset travel time is
already an Abelian group, *g*1=*T*. We apply a shear by nearest neighbor
interpolation along the
slowness squared axis to transform into the Dix's parameter, *g*2=*T*.*V*_{nmo}^{2}.
The results are shown in Figure 2,
where the waterbottom
multiples and peglegs lie evenly spaced along parallel straight lines in
the Abelian group plane.
The events in the velocity space span a large area because
of the insensitivity of velocity transformation.

Figure 2

The next most important property of
the Abelian group is *closure*. Closure means
that any linear combination of different Abelian group attributes also
forms an Abelian group. If we have a set of vectors *v*, which forms
such a group, then for any constant matrix, *M*, the set of vectors
*M*.*v*, also forms such a group. We introduce a shear into the
(*g*1,*g*2) coordinate frame in Figure 2 by replacing
(*T*.*V*_{nmo}^{2}) with the linear combination (*T*.*V*_{nmo}^{2} - *T*.*V*_{water}^{2})
since we are concerned with the water-bottom multiples and their peglegs
(or some other lower-bounding velocity
if other intrabed multiples are concerned).
The slope of the waterbottom multiples and peglegs - *V*_{water}^{2}
in (*g*1,*g*2) plane
can either be known from field water velocity measurement
or from looking at the Hilbert envelope of the data,
and then finding the shear
on the data (i. e., by varying *V*_{water}^{2}) that maximizes the sum of the
envelope energy over axis *T*.
Then the
waterbottom multiples and peglegs are steered straight down the *T* axis.
Thus in this domain, predictive multiple attenuation operators will work well.
This process yields the new Abelian group attributes:

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

11/18/1997