Estimating the data covariance

If we change sides in equation equ9 and rewrite it in the more standard form:

$$\bf d={\bf A} \bf \hat{d}, \EQNLABEL{equ10}$$ (12)

then equation equ10 looks similar to the linear relation equ-forward that relates a data vector to a model. Given that ${\bf A}$ is essentially an AMO matrix, then all its elements are positive. This is also evident since ${\bf A}$ represents a cross product matrix.

To estimate an approximate inverse for ${\bf A}$ we apply the same normalization techniques in computing its inner product entries, which are, AMO transformation from a given input geometry to another. This normalization makes the cross-product matrix unit-less. Therefore when approximating ${\bf A^{-1}}$ by its transpose we avoid the ambiguity of scaling this adjoint. Moreover, since ${\bf A}$ is hermitian, then it is equal to its transpose.

We conclude that ${\bf A}$ is itself a data covariance matrix. It represents an equalization filter that measures the interdependencies among the data elements and corrects the imaging operator for the effects of fold variations.