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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 . For each value of , we obtain the
field by solving the system:
This system is solved using Levinson algorithm, since the matrix *L*^{T}*WL*
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*(*t*_{0},*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 , we multiply it by the matrix
*L* (depending on ):

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

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Stanford Exploration Project

1/13/1998