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Restrictions

Up to now, I only spoke of the overdetermined problem, which implies the inversion of the matrix LTL. Things are different in the underdetermined case, where we have to invert LLT. Effectively, this matrix is not Toeplitz, even for a regular sampling in offset. The expression of a component of LLT is:

With a regular sampling in offset, supposing h0=0, this term becomes:

Consequently, (LLT)ik does not depend only on , and the structure of the matrix LLT 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 LT: (LLT)-1L in the overdetermined case, (LTL)-1LT 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