%In Chapter~\ref{chap:reg} and Appendix~\ref{chap:nonstat} I introduced
%several different methods to characterize the covariance. 
%In each case I presented  a single answer to the interpolation or
%tomography problem.  
\par
As discussed in Chapter~\ref{chap:reg},  geophysicists and geostatisticians have
very similar strategies.
Each use the known data to characterize the model's covariance.  Geophysicists
often characterize the covariance through Prediction Error Filters (PEFs)
\cite{gee}.  Geostatisticians build variograms  from the known data
to represent the model's covariance \cite{geostat}.  
Once each has some measure of the model covariance they attempt to fill
in the missing data with the goal of creating a model whose covariance
is equivalent to the covariance of the known data.
Both of these approaches are in some ways exactly what we are looking for:
given a problem, give me `the answer'.  
\par
The single solution approach
however has a couple significant drawbacks.  First, the solution tends to 
have low spatial frequency. Second, it does not provide information
on model variability or provide error bars on our model estimate.
Geostatisticians  have these
abilities  in their repertoire through
what they refer to as `multiple realizations'.
They introduce a random component, based on
properties (such as variance) of the data,   to their estimation procedure.  
Each realization's frequency content is more accurate and by
comparing and contrasting  the equi-probable realizations,
model variability can be assessed.
\par
In this appendix I present a method to achieve the same goal
using a formulation that better fits into geophysical techniques.  
I modify the model styling goal,
replacing the zero vector with a random vector.  I show  how
the resulting models have a more pleasing texture and can
provide information on variability.
