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Up to now, I only spoke of the *overdetermined* problem, which implies
the inversion of the matrix *L*^{T}*L*. Things are different in the
*underdetermined* case, where we have to invert *LL*^{T}. Effectively, this
matrix is not Toeplitz, even for a regular sampling in offset. The expression
of a component of *LL*^{T} is:
With a regular sampling in offset, supposing *h*_{0}=0,
this term becomes:
Consequently, (*LL*^{T})_{ik} does not depend only on , and the
structure of the matrix *LL*^{T} is not Toeplitz. So, it is not possible
to use Levinson algorithm in the underdetermined transformation
, where we have more parameters
*p* than offsets *h*.
We can reason similarly for the transform
. Suppose that, rather than using simply the
modeling matrix *L*, we want to compute the least-squares inverse of *L*^{T}:
(*LL*^{T})^{-1}*L* in the overdetermined case, (*L*^{T}*L*)^{-1}*L*^{T} in the
underdetermined case. Thus, Levinson algorithm could only be applied in
the underdetermined case.

** Next:** Summary of the transforms
** Up:** EXPRESSIONS OF THE PARABOLIC
** Previous:** Inverse transformation
Stanford Exploration Project

1/13/1998