Inverse transformation

In the case of hyperbolic transforms, the inversion (5) is expensive. However, for parabolic as for slant-stack transforms, the computation of (L^TL)^-1L^T is easy once we transform the initial data into the frequency domain. First, the size of the matrix L is consequently reduced: we have to invert matrices of size $n_p\times n_p$ (n_p is the number of p-parameters used in the computation).

What is more important is the Toeplitz structure of the matrix L^TL. 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 $\delta p$), (L^TL)_ik is independent of (i-k):

$$(L^TL)_{ik}=\sum_{h}\exp(j\omega(i-k)(\delta p)h^2) \;.$$

This diagonal invariance defines the Toeplitz structure of the matrix L^TL.

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:

$$U(\omega,.)=(L^TWL)^{-1}.L^TWd(\omega,.) \;.$$

But, if W=diag(w(h)), we see that:

$$(L^TWL)_{ik}=\sum_{h}w(h)\exp(j\omega(i-k)(\delta p)h^2) \;,$$

and the diagonal invariance, so the Toeplitz structure, are preserved.

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 $d(t,h)\rightarrow U(t_0,p)$ turns out to be very easy in the frequency domain. For each frequency, if we use n_p parameters p, it requires O(n_p²) operations, compared to O(n_p³) operations for more general methods, like Cholevski or SVD decompositions, or the conjugate-gradient method.