Both the geostatistical and geophysical approach give poor results because
we are ignoring a basic tenant
that both methods are built on: stationarity.    An underlying assumption of
both methods is that the statistics of the data do not vary with location.
If we look at the covariance at four different regions in the
data, Figure~\ref{fig:covar-change}, we can see that covariance changes
dramatically throughout the model.
The top left covariance is
taken from the flat structure at the top of the model, and shows
a strong horizontal trend. The top right represents
the covariance along the upper portion of the anticline and has a slight
dip down to the left.  The bottom right is the covariance
from the lower portion of the
anticline and captures the sharper dip down-left in the region.  The final
panel, the bottom-right represents the area below the unconformity and
captures the up-dipping structure. 

\plot{covar-change}{width=6.0in,height=3.0in}{The covariance at four
different regions of our model (left panel of Figure~\ref{fig:well-logs}).
The top left is above the upper unconformity; top right, the upper portion
of the anticline; bottom left, lower portion of the anticline; and bottom
right, below the lower anticline.}

\subsection{Patching}
A common solution to 
this non-stationarity is to break the problem into patches  
(geophysics terminology)
\cite{Claerbout.sep.73.391}
or `distinct sub zones' (geostatistics terminology).  
We define regions where the assumption of
stationarity is valid, and apply the given technique in the region.  We
then recombine the sub-regions into our final model.
Figure~\ref{fig:patch} shows the result of defining our model
space into the four different regions 
of Figure~\ref{fig:covar-change} and then applying the geophysical and
geostatistical approach. Note that our image is significantly improved
over the single covariance function approach.  The {\it range} description
for the geostatistical approach is poor, the images 
are still of significantly lower frequency than the known model
(multiple realizations, a geophysics equivalent of which is presented
in Appendix~\ref{chap:random}, can provide a higher frequency answer), and we
still haven't done a very good job within the anticline.



\plot{patch}{width=6.0in,height=3.0in}{The result of breaking
the problem into four patches  and solving them independently.
Left panel is the result of applying the geophysical method,
the right geostatistics.}

%\subsection{Non-stationary filtering}
%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. 
%\par
%An alternative  approach,
%proven to be more effective when dips change 
%quickly, is  to estimate PEFs in micro patches with
%a space varying PEF \cite{Crawley.segab.99}. The space varying estimation uses 
%the same initial fitting goals (\ref{eq:pef}) as the stationary approach
%but ${\bf a}$ is now many filters (taken to the extreme a filter at
%every model point) each operating in a different portion of the model.
%With so many filters, and therefore filter coefficients, the problem
%goes from being over-determined to under-determined.
%As a result we need to add a regularization equation to 
%our filter estimation fitting goals,
%\beqa
%\zero \pox \bf M \bf a \\
%\zero \pox \epsilon \bf F \bf a  \nonumber
%\eeqa
%where the regularization operator $\bf F$ smoothes the filter coefficients 
\cite{Clapp.sep.100.bob3}.
%By estimating and regularizing with the  space varying PEF
%we can almost perfectly recover our initial model
%(Figure~\ref{fig:gp-continuous}).
%We have significantly higher frequency content than the patching approach
%and have done a good job recovering the quickly changing slopes in the anticline.
%
%\sideplot{gp-continuous}{width=3.0in,height=3.0in}{The result of finding
%non-stationary PEF using the correct model and then applying it to
%the missing data problem. The returned image is almost exactly correct.}

