To give you an example of the ability of this operator to steer a solutions,
lets look at a simple missing data problem.
We will start with a slice form a rather complex synthetic velocity model 
(Figure~\ref{fig:qdome-geophysics}.  that has a dome -feature overlaying
an angular unconformity.
We will chose, some representative velocity boundaries,
calculate their dip at each point along the boundary, and then smoothly spread
this dip information to every point in the model space, Figure~\ref{fig:qdome-geophysics}.
We simulate nine well logs along the survey 
by subsampling the velocity field.
Next we set up our fitting goals for our problem.
We want to honor our data, so we won't allow the model to change at known
locations ($\bold J$ will  be a diagonal masking operator with 
zeros at known  model locations, ones otherwise), and we want to regularize
with our steering filter operator ($\reg$),

\plot{qdome-geophysics}{width=6.0in}{Left panel represent the input velocity
field with some representative reflectors highlighted.  The center panel
is the result of calculating the dip of these reflectors and smoothing
into the entire model space.  The right panel is the input data to our
interpolation problem.}

\beqa
\data &\approx& (\iop - \bold J) \model \\ \nonumber
\zero &\approx& \reg \model .
\eeqa

After 120 iterations, you can see that we did an excellent job recovering
our initial velocity model (Figure~\ref{fig:sds}), 
for comparison I have included the result
of using the Laplacian as our operator.

By reformulating the problem into 1-D using the helix
we can can change our regularized 
interpolation problem, into a preconditioned problem,
\beqa
\data &\approx& (\iop - \bold J) \prec \pvar \\ \nonumber
\zero &\approx& \pvar .
\eeqa
Applying this new set of equations to the same problem, we get
convergence, to approximately the same model in 12, rather than 120 
iterations, a two orders of magnitude speed up!
