With no other information the Laplacian might be the best regularization 
operator that we could use. But if we know something else
about out model, can we do better?  According to \longcite{tarantola-ipt}
we should be using the inverse model covariance for our
regularization operator.
For this simple example where we know the `correct' answer we could
calculate the model covariance matrix.  In a real problem 
we would not have this luxury.  A more reasonable goal is to attempt
to estimate the covariance of the model as a whole. We can obtain an
estimate of the model covariance $C(\bf x)$ by taking the
auto correlation of the model and subtracting the mean of the data or by
\beq
C({\bf x})= < y_i y_j > - <y_i> < y_j>
\eeq
where  $y$  is the data points, and
$< >$ is an average of all of the points separated
by the  vector $\bf x$.

%{1 \over N({\bf x})}
%\sum_{(i,j)| xdist={\bf x}}
%d_i d_j-m_{-{\bf x}} m_{+{\bf x}}
%\eeq
%where  we have the data values $d_1 ... d_n$.
%We are summing the   $N({\bf x})$  points that
%have $\bf x$ separation and subtract the product
%of  
%the mean ($m$) of data points  $+{\bf x}$ and $-{\bf x}$
%away from a given data point \cite{geostat}.
Figure~\ref{fig:model-covar} shows the 2-D auto-correlation of the left
panel in
Figure~\ref{fig:well-logs}.
Geophysicists and geostatisticians both  use the model covariance,
although in different ways, to solve the missing
data problem.  
\par
\sideplot{model-covar}{width=3.0in,height=3.0in}
{Spatial autocorrelation matrix for the velocity model in Figure~\ref{fig:well-logs}.
Note that the dip below the lower unconformity dominates
the covariance calculation.}

\subsection{Geophysical approach}
\sepinput{geophysics}

\subsection{Geostatistical approach}
\sepinput{geostat}
