For the data-space inverse solution, the input is
first filtered with the inverse of the operator **A**.
The main challenge is then to solve for this inverse and for that we start
by writing the solution for *m* from equation (3) in terms of **A** as

(8) |

(9) |

where is the filtered input given by the substitution:

(10) |

Solving for , we then need to compute the inverse of from the system of equations:

(11) |

Once the inverse of is estimated to yield the filtered data , we merely evaluate to get the solution for the original problem.

Note that after filtering,
one can use any imaging operator *L*^{T} to invert for *m*.
The new input is well suited to prestack imaging based on any wave
equation operator.
The role of the equalization filter
was to correct the imaging operator for the interdependencies
between data elements.

Comparing the fold-normalization technique proposed by Chemingui and Biondi
(1996) to the least-squares solution in (9),
the inverse of the cross product matrix (*LL*^{T})
was approximated by a diagonal matrix in the normalization solution.
The diagonal elements in the normalization solution
were heuristically derived to be proportional to the inverse of the
fold coverage at the output bin that corresponds to
the data trace. The fold-normalization procedure,
therefore, provides an approximate
solution to the problem of varying fold coverage. Nevertheless, experience
has proved it can provide good results.

11/11/1997