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## Modeling operation

The modeling equation is similar to Thorson's equation for hyperbolic transforms. It expresses the NMO-corrected data d(t,h) as a function of the field U in the t0-p domain:
 (3)
whereas, for Thorson's equation, the term has to be replaced by  . Similarly, it can be compared to the slant-stack modeling:

Equation (3) will transform a spike in the t0-p domain into a parabola (or straight line if p=0) in the time-offset domain.

I symbolize equation (3) as: d=L.U. Equation (3) can be expressed in the frequency domain. For the transform, Kostov (1989) showed that:

For the parabolic transform, we get:
 (4)
So, for a fixed , an element of the matrix L is:

Stacking along parabolas actually corresponds to computing U=LT.d (as Thorson pointed out in the case of hyperbolic stacking). We will transform the data d(t,h) into the t0-p domain by using the least-squares inverse of L, i.e. (LTL)-1LT in the overdetermined case (or LT(LLT)-1 in the underdetermined case):
 (5)

Next: Inverse transformation Up: EXPRESSIONS OF THE PARABOLIC Previous: EXPRESSIONS OF THE PARABOLIC
Stanford Exploration Project
1/13/1998