Column scaling: data normalization

Recall that each column of ${\bf F}$ corresponds to an impulse response. We therefore apply column scaling to imaging operators implemented as push operators. The system we solve is then

$$\bold m = \bold F \bold C^{-1} \bold d, \EQNLABEL{equ-dat}$$ (51)

where the sum of the elements of each column are on the diagonal of ${\bf C}$.

In equation equ-dat the imaging operator is applied after the data have been normalized by ${\bf C^{-1}}$ and, consequently, we will refer to this normalization as data normalization. Again we notice that ${\bf C^{-1}}$ has the inverse units of ${\bf F}$ making the output image unit-less.

Similarly to the imaging fold defined earlier, ${\bf C^{-1}}$ is normalization by the coverage of the modeling operator ${\bf L}$(where $\bf F = \bf L^{-1}$). We will refer to this coverage as the modeling fold.