We need to transform the NMO-corrected data to the t0-p domain, t0 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 t0-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 t0-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.