The geophysicist attempts to characterize the model covariance 
through a Prediction Error Filter (PEF).  The geophysicist notes that 
if we solve
\beq
\bold M \bold a \approx \zero \label{eq:pef} ,
\eeq
where $\bold M$ is convolution with a field that has the same properties
as the model, and $\bold a$ is our PEF,
the output of this convolution is white \cite{pvi}. Therefore $\bf a$ must
have  
the inverse spectrum
of  the model. This is only true if we have chosen 
a sufficient shape and size for $\bf a$.
By changing the shape of $\bf a$ we can
control how many 
and  which dips 
of the model   we see.
If we construct  $\bf a$ to find dips at all possible
angles and then
apply (\ref{eq:pef}) we
get  Figure~\ref{fig:gp-mis} as the impulse response of our filter.
%\footnote{In truth this is the sum of the impulse responses of ${\bf a^{-1}}$
%and ${\bf a^{-1'}}$. Note how the impulse response
%is approximately equal to our model covariance,
%Figure~\ref{fig:model-covar} If we use ${\bf a}$ as a regularization operator 
%we will get an approximately equivalent result.}.
The PEF captured the prominent dip going up at approximately 15 degrees
and also a minor dip going down at approximately 30 degrees.
The right panel of Figure~\ref{fig:gp-mis} shows the result of applying
the PEF to our missing data problem.  We have done a substantially better
job filling in the missing data compared to our Laplacian result,
Figure~\ref{fig:qdome-lap}, but its still far from ideal.
The filter has introduced both dips
at every location and as a result we have a  model that is    
unreasonable.
\plot{gp-mis}{width=6.0in,height=3.0in}
{
The left panel shows the impulse response  of the PEF found from
the velocity field (left panel of Figure~\ref{fig:well-logs}).
The right panel shows the result of applying the PEF
to the missing
data problem ( the input being the right side of Figure~\ref{fig:well-logs}).}

