Review of regularized inversion

Iterative least-squares inversion can be expressed simply as the conjugate-gradient minimization of this objective function:

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

where boldL is a linear modeling operator, boldd is the data, and boldm is the model. In this paper, L is the adjoint of the downward continuation migration operator explained by Prucha et al. (1999b). This migration algorithm produces a model with the axes depth (z), common reflection point (CRP), and offset ray parameter (p_h). Offset ray parameter is related to the reflection angle $\theta$ (where $\theta$is half of the opening angle between incident and reflected rays) by:

 

$$p_{h}\ =\ \frac{2\sin \theta \cos\phi}{V(z,CRP)}.$$ (2)

where $\phi$ is the local dip and V(z,CRP) is the local velocity at the reflection point.

The minimization can be expressed more concisely as a fitting goal:

$${\bf 0}\ \approx \ {\bf L m}\ -\ {\bf d}.$$ (3)

However, many issues exist, including poor illumination, that make this iterative inversion likely to have a large null space. Any noise that exists within that null space will not be constrained during the inversion and 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 as we minimize the first one:

$$\begin{eqnarray} {\bf 0} &\approx&{\bf L m}\ -\ {\bf d} \\ {\bf 0} &\approx&\epsilon{\bf A m}. \nonumber\end{eqnarray}$$ (4)

The first expression in (4) 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 (4) 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:

$$\begin{eqnarray} {\bf 0} &\approx&{\bf LA^{-1}p\ -\ d } \\ {\bf 0} &\approx&\epsilon{\bf p}. \nonumber\end{eqnarray}$$ (5)

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