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:  

$$d(t,h)=\sum_{t_0}\sum_p U(t_0,p)\delta[t_0-(t-ph^2)]$$ (3)

whereas, for Thorson's equation, the $\delta$ term has to be replaced by $\delta\!~[~t_0~-~\sqrt{t^2-p^2h^2}~]$ . Similarly, it can be compared to the slant-stack modeling:

$$d(t,h)=\sum_{\tau}\sum_p U(\tau,p)\delta[\tau-(t-ph)]\;.$$

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 $\tau-p$ transform, Kostov (1989) showed that:

$$d(\omega,h)=\sum_p U(\omega,p)e^{-j\omega ph} \;.$$

For the parabolic transform, we get:

$$d(\omega,h)=\sum_p U(\omega,p)e^{-j\omega ph^2} \;$$ (4)

So, for a fixed $\omega$, an element of the matrix L is:

$$L_{ik}=\exp(-j\omega p_kh_i^2) \;.$$

Stacking along parabolas actually corresponds to computing U=L^T.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. (L^TL)^-1L^T in the overdetermined case (or L^T(LL^T)^-1 in the underdetermined case):  

$$U(\omega,.)=(L^TL)^{-1}.L^Td(\omega,.) \mbox{\hspace{1.0cm}($\omega$\space is fixed)}$$ (5)