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Review of regularized inversion

The vast size of the seismic imaging problems makes performing a direct inversion impossible with today's computer power, even if we are only dealing with a 2-D seismic line. Fortunately, we can closely approximate a direct inverse with iterative techniques. In particular, we can approximate a least-squares inversion with the conjugate-gradient minimization of this objective function:

Q({\bf m})\ =\ \vert\vert{\bf L m}\ -\ {\bf d}\vert\vert^{2} \end{displaymath} (1)

where boldL is a linear modeling operator, boldd is the data, and boldm is the model. This minimization can be expressed more concisely as a fitting goal:

{\bf 0}\ \approx \ {\bf L m}\ -\ {\bf d}.\end{displaymath} (2)
However, for the seismic imaging problem, this inversion can have a large null space, due in part to poor illumination. Any noise that exists within that null space can grow with each iteration until the problem becomes unstable. Fortunately, we can stabilize this problem with regularization Tikhonov and Arsenin (1977). The regularization adds a second fitting goal that we are minimizing at the same time:

{\bf 0} &\approx& {\bf L m}\ -\ {\bf d}
\\ {\bf 0} &\approx& \epsilon {\bf A m}. \nonumber\end{eqnarray} (3)

The first expression is the ``data fitting goal,'' meaning that it is responsible for making a model that is consistent with the data. The second expression is the ``model styling goal,'' meaning that it allows us to impose some idea of what the model should look like using the regularization operator ${\bf A}$. The strength of the regularization is controlled by the regularization parameter $\epsilon$.

Unfortunately, the inversion process described by fitting goals (3) can take many iterations to produce a satisfactory result. We can reduce the necessary number of iterations by making the problem a preconditioned one. We use the preconditioning transformation ${\bf m = A^{-1}p}$ Fomel et al. (1997); Fomel and Claerbout (2003) to give us these fitting goals:

{\bf 0} &\approx& {\bf LA^{-1}p\ -\ d }
\\ {\bf 0} &\approx& \epsilon {\bf p}. \nonumber\end{eqnarray} (4)

${\bf A^{-1}}$ is obtained by mapping the multi-dimensional regularization operator ${\bf A}$ to helical space and applying polynomial division Claerbout (1998). This process is called Regularized Inversion with model Preconditioning (RIP).