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The method of multiples elimination by parabolic transforms was first
proposed by Hampson (1986 a). It uses the fact that, after NMO correction
of a CMP gather, the shape of the multiples is approximately parabolic,
while the primaries are more or less horizontal. To suppress the multiples,
the process will thus separate the parabolic events from the horizontal
events, characterizing them by their curvature parameters. Yilmaz (1989)
proposed a similar method, to be applied on the original data
(not NMO-corrected), after stretching the time axis.
We need to transform the NMO-corrected data to the *t*_{0}-*p* domain, *t*_{0}
being the 0-offset time, and *p* the curvature of the parabolas. Actually,
the basic transform is a modeling process, transforming a spike
in the *t*_{0}-*p* domain into a parabola in the time-offset domain. In this
sense, it is similar to hyperbolic or slant-stack transforms, where
a spike is respectively transformed into an hyperbola or a straight line
(Thorson, 1984; Kostov, 1989). The modeling can be easily expressed in the
frequency domain, as Kostov (1989) already pointed out.

However, the inverse transformation (time-offset to *t*_{0}-*p*) needs the
least-squares inverse of this modeling operator, for each frequency.
Hampson (1986 b) did not indeed mention how he performed this inversion;
he just suggested that the size of the matrices involved allow the use of
``array processor'' routines. Yilmaz (1989) used SVD decompositions, which can
be prohibitively expensive, especially since we have to perform an inversion
for each frequency!

But, as Kostov (1989) already showed in the case of the slant-stack transform,
I will show that the matrix being inverted in this computation has
indeed a Toeplitz structure. Thus, the inversion can be performed very
easily with Levinson algorithm, and the cost of the process is substantially
reduced. Moreover, we can also include offset-dependent weighting without
losing this structure; these properties are independent of the space sampling,
may it be regular or irregular. For these reasons, this method,
already attractive for its efficiency, should be even more appealing and
used on a large scale.

This paper will essentially focus on the inversion operation. I will first
recall the philosophy of the multiples elimination process, which explains
the use of parabolic transforms. Then, I will define the forward and inverse
parabolic transforms, and show how the Toeplitz structure appears and can be
used. Finally, I will use real data, first to show the efficiency of the
multiples elimination process for water-bottom multiples and peglegs. I also
extend the applications to interpolation problems, where I will take advantage
of the Toeplitz structure to use an offset-dependent weighting on
irregularly sampled data.

** Next:** WHY PARABOLIC TRANSFORMS?
** Up:** Darche: Fast parabolic transforms
** Previous:** Darche: Fast parabolic transforms
Stanford Exploration Project

1/13/1998