Regularized linear problems can be written as minimizing the quadratic
function
\beq
Q(m) = \|d- Lm\|^2 + \epsilon^2 \|Am\|^2
\eeq
or in terms of  fitting goals,
\beqa
\data &\approx& \bf L \bf m \\ 
\label{eq:geophysics} 
\nonumber
\zero &\approx& \epsilon  \reg \bf m 
,
\eeqa
where $\data$ is our data, $\bf L$ is our modeling operator,
$\reg$ is our regularization operator and we are inverting
for  a model $\bf m$.
For the purpose of this appendix I will refer to the first
goal as the {\it model fitting goal} and the second as
the {\it model styling goal}.
Normally we think of {\it model fitting goal } as describing
the physics of the problem.  The {\it model styling goal} is suppose
to provide information about the model character.  Ideally 
$\reg$ should be the inverse model covariance.   In practice
we don't have the model covariance so we attempt  to approximate it through
another operator.   At SEP the regularization operator is  anything from:
\begin{description}
\item [laplacian or gradient]  a simple operator that assume nothing about the model
\item [Prediction Error Filter (PEF)]  a stationary operator  estimated from known portions of the model or  some field with same properties as the model \cite{gee}
\item [steering filter] a non-stationary operator built from minimal information about the model \cite{Clapp.sep.95.bob1}
\item [non-stationary PEF] a non-stationary operator built from a field with the same properties as the model \cite{Crawley.sep.104}
\end{description}

A problem with at least the first three of these approximations is
that while they approximate the model covariance, they
have little concept of model variance.  As a result our models tend
to have the wrong {\it character}.

