%When building a steering filter, we want to create a filter
%that destroys a given slope {\it p}. Further, we would like to
%control the bandwidth response of
%filters oriented at different
%slopes.  We can achieve both these goals by constructing
%a triangle centered at the appropriate slope (Figure~\ref{fig:steering}).
%Every grid
%cell center which the triangle passes through is assigned a negative value
%proportional to the height of the triangle at that location.
%
%
%The wider the triangle base, the less  {\it anisotropic}
%our smoother becomes  (Figure~\ref{fig:width}).  By decreasing the sum
%of the coefficients 
%we can increase our decrease the {\it range}.
%
%\sideplot[t]{steering}{height=2.5in}{A finite-difference star for a monoplane rejection filter.
%The left column contains a `1'.  The right column contains
%samples off a triangle.
%The desired
%slope is represented by {\it p}, the smaller {\it w} the more precise
%the dip smoothed, and the larger {\it h} the bigger the
%area the smoother acts on.}
%
%
%For example, let's return to the missing data problem.
If we  assume
that velocity follows
structure and we have some guesses at reflector position, we
can use this information to build our steering filters.
For this problem we will assume that we have the location
of four reflectors, one above the top unconformity, two between the 
unconformities and  one below the lower unconformity 
(left-panel of Figure~\ref{fig:qdome-refs}).
If we  interpolate these dips to our entire model space we have all
we need to construct a space-varying operator composed of steering filters.
If we use this operator as our regularizer, we get Figure~\ref{fig:qdome-reg-cont}
as our interpolation result.  The steering filters  did a significantly
better job than the isotropic regularizer.
%It
%didn't do as good of job as characterizing the model covariance as
%our space-varying PEF, Figure~\ref{fig:gp-continuous}, but it
%assumed a much more reasonable set of known information.
%With more information about our model (more reflectors, an idea on
%the level of {\it anisotropy} in different portions of the model, etc.) we
%could do even better by using some of the other adjustable
%parameters available when constructing the steering filter operator.
%
\plot{qdome-refs}{width=6.0in,height=3.0in}{Left panel
are four reflectors chosen to represent our {\it a priori} information.
The right panel is interpolated slope calculated from the reflectors that
will form the basis of our steering filter.}


\sideplot{qdome-reg-cont}{width=3.0in,height=3.0in}{The result
of using our steering filter operator as regularizer to the
missing data problem.}




%The number of adjustable parameters in the filter construction
%can seem daunting but is what enables almost any covariance
%function to be described by steering filters.
%In certain regions of the model you might feel that
%that  covariance function is much more isotropic.
%In these regions you  could consider making your triangle
%bigger to  smooth your filter coefficients over a wider angle {\it range}, while
%keeping it small in areas where the covariance is much
%more {\it anisotropic}.  The sum of the
%non-zero lag coefficients opens up another intriguing freedom.  As
%Figure~\ref{fig:distance} shows, when the sum of the non-zero lag coefficients
%gets close to $-1$, the area over which the smoother operates increases
%greatly.  This is similar to increasing the $\epsilon$ value over
%only a portion of your model space.  This enable steering filters
%to handle a model space where the sill difference is variable.
%
%\plot{width}{width=6.0in,height=3.3in}{The impulse response of the smoothing filter as
%function of the triangle base. Note the wider the base, the less precise
%the dip smoothing. }
%
%\plot{distance}{width=6.0in,height=3.3in}{The impulse response of the smoothing filter as
%the sum of the non-zero lag coefficients get closer to 1.}
%
%
%
%%%%%%%%%
