\def\figdir{./Figs}
%A continuously varying PEF does an excellent job describing our
%model covariance and solving our missing data problem but is
%often impractical. The geophysicist's approach assumed
%that we had a field with the same statistical properties
%as our model space and this is rarely the case.
%\par
If we reexamine our desired velocity function it is
apparent that the covariance function varies within at
least two of our  four patches (we can also
see this in the covariance function of patch 2 and 4 of Figure~\ref{fig:covar-change}).  Therefore it follows that;
we should get a better image by making smaller and smaller patches.
\longcite{Crawley.segab.98}
showed that this true when solving a data interpolation problem.
Unfortunately we can only make our patch size so small before
we can't generate sufficient statistics to find our PEF or
solve the krigging equations. 
In addition, our geophysical solution relies on having a field
with the same statistical properties as our model, this is
often not the case.
Often what we have is what  geostatisticians refer to as `soft data'.
Soft data has {\it generally} the same properties of the
variable of interest but it is often in incompatible form,
a classic example of this is tying well measurements to
flow simulation results.
Geostatistics
is generally better
suited for combining mixed and limited data. Krigging requires
us to provide only a variogram for each sub-zone.  
On the other hand, the geostatistical approach is not well suited
for fitting into an iterative optimization problem.  Krigging
wants the physics of the problem to be describable by a known function that
would then form the basis for a space varying mean for the krigging 
problem \cite{geostat}.
For problems like tomography this is an unacceptable requirement.
\par
In geophysics there have been attempts to combine different information
sources \cite{SEG-1994-1338},%,SEG-1996-0711} , 
but with limited success.  Generally, our inversion problems are
too large to use classical hard constraint mechanisms \cite{opbook}.
What we need is a method to construct a space-varying filter
that does a good job describing our model covariance, but
can be obtained from limited and disparate information sources.
\par
If we look at 
regions  two and four in
Figure~\ref{fig:covar-change} we can see that when our stationary assumption
is  valid the  covariance matrix is fairly
simple. We have a primary trend oriented along the dip of the
velocity field that slowly dies out and a ringing effect due
to the sinusoidal nature of our model.
We would like to come up with a way to emulate the primary trend of
the covariance matrix through minimal information.
\par
To do this it is important to remember that 
our regularization operator should have the inverse spectrum
of the covariance matrix.  Therefore if the covariance function is primarily
a dipping event, our regularization operator should be destroying
that dip.
Claerbout \shortcite{Claerbout.sep.65.229,pvi} showed how to estimate 
the primary dip in a region and how to construct a filter
that could destroy that dip.  These small filters, which I refer
to as {\it dip penalty filters}  can be as simple as a two or
three point filter, Figure~\ref{fig:small-filter}.
A dip penalty filter consists of a fixed `1' and one or more coefficients
in the next column.  The location of the filter coefficients in the
second column determines the dip that the  filter will destroy.
Figure~\ref{fig:small-response} is the inverse impulse response of
Figure~\ref{fig:small-filter}. Note how the general orientation
of the impulse response is approximately the same as
the covariance function below the lower unconformity, but the
{\it anisotropy} and {\it range} are not correct.  In the next section I will
discuss how we can also control these parameters with dip penalty
filters.

\activesideplot{small-filter}{height=1.0in}{NR}{A dip penalty filter
oriented approximately 22.5 degrees.}
\activesideplot{small-response}{height=3.0in,width=3.0in}{ER}{The result of applying
$\frac{1}{\bf A \bf A'}$ where $\bf A$ is the filter in
Figure~\ref{fig:small-filter}.}

