What is more important is the Toeplitz structure of the matrix *L*^{T}*L*.
Kostov (1989) showed this property in the case of the slant-stack transform.
This property still holds for the parabolic transform. Effectively,
assuming that the *p*-parameters are regularly sampled (with a step
), (*L*^{T}*L*)_{ik} is independent of (*i*-*k*):

Moreover, introducing an offset-dependent weighting does not perturb this
Toeplitz structure. Effectively, this is equivalent to using a diagonal
positive matrix *W* to compute the transform:

Notice also that this Toeplitz structure holds for any kind
of spatial sampling: particularly, it is true for any *irregular*
sampling, for example if some traces are missing. Finally, we
can even choose a frequency-dependent weighting, since the inversions
are performed separately at each frequency. For example, if the far offsets
are not sufficiently muted after NMO correction, their low-frequency
content due to stretching could influence the inversion process. Thus, it can
seem reasonable to give low weights at low frequencies to these
far offsets.

Consequently, using Levinson algorithm, the transform turns out to be very easy in the frequency domain. For each
frequency, if we use *n*_{p} parameters *p*, it requires *O*(*n*_{p}^{2}) operations,
compared to *O*(*n*_{p}^{3}) operations for more general methods, like Cholevski
or SVD decompositions, or the conjugate-gradient method.

1/13/1998