Data equalization

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  

$$m=L^T {\bf A}^{-1} d$$ (8)

Then we change the problem formulation variable d to a new variable $\hat{d}$and recast the problem as  

$$m=L^T \hat{d}$$ (9)

where $\hat{d}$ is the filtered input given by the substitution:  

$$\hat{d}={\bf A}^{-1}d$$ (10)

Solving for $\hat{d}$, we then need to compute the inverse of ${\bf A}$ from the system of equations:  

$$d={\bf A} \hat{d}$$ (11)

Once the inverse of ${\bf A}$ is estimated to yield the filtered data $\hat{d}$, we merely evaluate $m=L^T \hat{d}$ 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.