Normalizing vs scaling of the adjoint

Imaging is quite often derived as the adjoint of modeling, where in the absence of explicit formulation for ${\bf F}$ we seek an approximate inverse for ${\bf L}$. Mathematically, this means that we approximate an inverse of a matrix of very high order by the transpose (Hilbert adjoint) of ${\bf L}$.

Claerbout 1998 points out that unless ${\bf L}$ has no physical units, the units of the transpose solution $\bf m_t = \bf L^T \bf d$ do not match those of $\bf m_t = \bf F \bf d$. Given the theoretical (least squares) solution $\bf m_{lsq} = (\bf L \bf L^T)^{-1} \bf F^T \bf d$, Claerbout suggests that the scaling units should be those of $(\bf L \bf L^T)^{-1}$. He proposes a diagonal weighting function suggested by Bill Symes (private communication) that makes the image $\bf m_t = \bf W^2 \bf L^T \bf d$, where we chose the weighting function to be  

$$\bold W^2 = {\bf diag} \left( {\bold L^{T} \bold d \over \bold L^{T} \bold L\bold L^T \bold d} \right),$$ (5)

which obviously has the correct physical units.

Comparing the normalized solution to the scaled adjoint, we see that the normalized solution is unit-less. It therefore avoids the ambiguity in guessing approximate weights. The normalized solution represents a ratio of two images where the reference image is the output of an input vector with all components being equal to one. This is equivalent to a calibration by a flat event response. Similar approaches might exist in practice some of them derived in heuristic ways, e.g., the DMO fold Slawson et al. (1995).

The normalization technique can also be applied to approximate inverses; i.e., to the adjoint. The normalized solution is again unit-less and represents a ratio of two approximate images. This will not make the final image exact, but, similarly to the case of normalized inverse, it calibrates the data for the effects of varying fold. In the next section we show how we can apply this concept to better approximate a pseudo-inverse and estimate a data covariance matrix.