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Review of regularized inversion

Least-squares inversion can be expressed simply as the minimization of this objective function:
\begin{displaymath}
min\{Q({\bf m})\ =\ \vert\vert{\bf L m}\ -\ {\bf d}\vert\vert^{2}\} \end{displaymath} (1)

where L is a linear modeling operator, d is the data, and m is the model. This minimization can be expressed more concisely as a fitting goal:
\begin{displaymath}
{\bf 0}\ \approx \ {\bf L m}\ -\ {\bf d}.\end{displaymath} (2)
However, in areas of poor illumination, this problem will have a large null space. The null space is partially caused by the fact that our survey can not have infinite extents and infinitely dense source and receiver grids. Any noise that exists within the null space can grow with each iteration until the problem becomes unstable. Fortunately, we can stabilize this problem with Tikhonov regularization Tikhonov and Arsenin (1977). The regularization adds a second fitting goal that we are minimizing:
   \begin{eqnarray}
{\bf 0} &\approx&{\bf L m}\ -\ {\bf d}
\\ {\bf 0} &\approx&\epsilon{\bf A m}. \nonumber\end{eqnarray} (3)
The first fitting goal is the ``data fitting goal,'' meaning that it is responsible for making a model that is consistent with the data. The second fitting goal 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 (2002) to give us these fitting goals:
   \begin{eqnarray}
{\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). After obtaining ${\bf p}$ from the fitting goals in (4), it is simple to transform back to ${\bf m}$. Now that our inversion is defined, we can take a closer look at $\epsilon$ and at the number of iterations (niter).


next up previous print clean
Next: The question of and niter Up: M. Clapp: The oddities Previous: Introduction
Stanford Exploration Project
7/8/2003