Data-space weighting functions

If the system of equations, ${\bf d}={\bf A}\, {\bf m}$,is underdetermined, then a standard approach is to find the solution with the minimum norm. For the L2 norm, this is the solution to  

$$\hat{\bf m}_{L2} ={\bf A}' \; \left( {\bf A} \, {\bf A}' \right)^{-1} \; {\bf d}.$$ (8)

As a corollary to the the methodology outlined above for creating model-space weighting functions, Claerbout (1998) suggests constructing diagonal approximations to ${\bf A}\, {\bf A}'$ by probing the operator with a reference data vector, ${\bf d}_{\rm ref}$. This gives data-space weighting functions of the form,  

$${\bf W}_{\rm d}^{2} = \frac{ {\rm\bf diag} ({\bf d}_{\rm ref... ...}_{\rm ref}) } \approx \left( {\bf A}\, {\bf A}' \right)^{-1},$$ (9)

which can be used to provide a direct approximation to the solution in equation (8),

$$\hat{\bf m}_{L2} \approx {\bf A}' \; {\bf W}_{\rm d}^2 \; {\bf d}.$$ (10)

Alternatively, we could use ${\bf W}_{\rm d}$ as a data-space preconditioning operator to help speed up the convergence of an iterative solver:  

$${\bf W}_{\rm d} \, {\bf d}={\bf W}_{\rm d} \, {\bf A}\, {\bf m}.$$ (11)