Two-step solution

The fundamental definition of A from equation equ6 allows analytic computations of its inner product elements. The challenge is then to solve for the inverse of A. First, we write the solution for ${\bf m}$ from equation equ3 in terms of A as

$$\bf m=\bf L^T {\bf A}^{-1} \bf d. \EQNLABEL{equ8}$$ (9)

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

$$\bf m=\bf L^T \bf \hat{d} \EQNLABEL{equ8}$$ (10)

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

$$\bf \hat{d}={\bf A}^{-1}\bf d \EQNLABEL{equ9}$$ (11)

Solving for $\bf \hat{d}$, we then need to compute the inverse of ${\bf A}$. This is essentially the first step of the solution, i.e., the data-equalization step. After filtering, we merely need to apply the imaging operator to the equalized data to obtain the final image. At this stage, any true-amplitude imaging process could be applied, e.g., prestack Kirchhoff migration.