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:
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):