Summary of the transforms

Starting from a CMP-gather, we apply a NMO correction with the velocity curve of the primaries, obtaining a data set d(t,h). Then, we apply a Fourier transform in the time direction to transform the data set to $d(\omega,h)$. For each value of $\omega$, we obtain the field $U(\omega,p)$ by solving the system:

$$(L^TWL)U(\omega,.)=L^TWd(\omega,.) \;.$$

This system is solved using Levinson algorithm, since the matrix L^TWL is Toeplitz. This property is independent of the weighting matrix W; it just needs to be diagonal. It is also true for any spatial sampling.

We get the field U(t₀,p) by applying the inverse Fourier in the time direction. After having filtered the field U (for multiple removal, for instance), we come back to the time-offset domain with the modeling operator L. To do so, we transform the field U to the frequency domain, and for each frequency $\omega$, we multiply it by the matrix L (depending on $\omega$):

$$d(\omega,.)=L.U(\omega,.)\;.$$

Finally, applying an inverse Fourier transform and an inverse NMO correction brings us back to the original time-offset domain.