The operators

In order to examine the results of preconditioned inversion, regularized inversion, and my proposed CIPR, I needed a problem that was easier to understand than that shown in Prucha and Biondi (2002). I am concerned with two issues: frequency content and solutions at every model point. To address the first issue, I chose to make my inversion operator a ``smoother'' that causes the model to have a higher frequency content than the data. This can be expressed as:

 

$${\bf d \approx Sm}$$ (1)

where ${\bf d}$ is the data, ${\bf m}$ is the model, and ${\bf S}$ is a smoothing operator that maps the average of 5 vertical points in the model to one point in the data. Since the model should be high frequency, the effects of the preconditioned inversion should be quite obvious.

Given such a simple inversion operator, creating a need for regularization or preconditioning requires that I cause the model created by inversion (fitting goal (1)) to have points that do not have solutions defined by the inversion operator. I chose to do this by introducing a masking operator ${\bf W}$.The combined operator ${\bf WS}$ will now have a null space where ${\bf W}\ =\ {\bf 0}$.This changes my fitting goal to:

 

$${\bf d \approx WSm} .$$ (2)

To interpolate the model in the areas affected by the null space, I add a second fitting goal to fitting goal (2):

$$\begin{eqnarray} {\bf d} &\approx& {\bf WSm} \\ {\bf 0} &\approx& \epsilon {\bf A m} \nonumber\end{eqnarray}$$ (3)

where the new operator, ${\bf A}$, is a regularization operator. I have chosen to make ${\bf A}$ a steering filter Clapp et al. (1997); Clapp (2001) generated as described in Prucha et al. (2000, 2001). Briefly, a steering filter consists of dip penalty filters at every model point, meaning that it is a non-stationary roughening operator that acts over short distances. To precondition this problem, I perform a change of variables to replace the model ${\bf m}$ with the preconditioned variable ${\bf p}$:

$${\bf m = A^{-1}p}.$$ (4)

Applying this to fitting goals (3) results in a new set of fitting goals:

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

The inverse of the steering filter (${\bf A^{-1}}$) is applied using the helix transform. The inverse operator will be a smoothing operator that will act over a much larger distance than ${\bf A}$.