Normalizing vs scaling of the adjoint

Imaging is 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 1999 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 the weighting function is  

$$\bold W^2 = {\bf diag} \left( {\bold L^{T} \bold d \over \bold L^{T} \bold L\bold L^T \bold d} \right),$$ (43)

which obviously has the correct physical units.

In contrast to the scaled adjoint, the normalized solution is unitless. It therefore avoids the ambiguity of guessing approximate weights. The model 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 the response of a flat event. Similar approaches might exist in practice, often derived in heuristic ways, e.g., the DMO fold Slawson et al. (1995).