\def\figdir{./Figs}
The missing data problem is probably the simplest to understand and
interpret results.  
We  begin by binning our data onto a regular mesh.
For $\bf L$ in fitting goals (\ref{eq:geophysics}) we will use a selector 
matrix $\bf J$,
which is `1' at locations where we have data and `0' at unknown locations.
For  this missing data problem 
I will use 
a day's worth of  data
collected by SeaBeam (Figure~\ref{fig:sea.init}), which measures
water depth under, and to the side, of a  ship \cite{gee}. 

\activesideplot{sea.init}{width=3.0in,height=3.0in}{ER}{Depth of the ocean under ship tracks.}

Figure~\ref{fig:sea.pef} shows the result of estimating a PEF from the
known data locations and then using it to interpolate the entire mesh.
Note how the solution has a lower spatial frequency as we move away from
the recorded data. In addition the original tracks of the ship are still
clearly visible.  

\activesideplot{sea.pef}{width=3.0in,height=3.0in}{ER}{Result of using a PEF to
interpolate Figure~\ref{fig:sea.init}, taken from GEE \cite{gee}.}

\par
If we look at a histograms of the known data and our estimated data we
can see the effect of the PEF.  The histogram of the known data has a
nice Gaussian shape.  The predicted
data is much less Gaussian with a much lower variance.   The problem
is how can we get the estimated model to have a  variance similar
to the known data?
\activesideplot{pef.histo}{width=2.0in,height=2.0in}{ER}{Histogram for the known data 
(solid lines) and the estimated data (`*').  Note the dissimilar
shapes.}
\par
Geostatisticians are confronted with the same problem. They can produce
smooth, low frequency models through krigging.  Unfortunately, often models
estimated from krigging prove to be poor representations of earth properties.
A classic examples is fluid flow where krigged models  tend to give inaccurate
predictions. Their solution
is to perform Gaussian simulation, rather than krigging, to
estimate the field \cite{geostat2}.
%is to  build a Cumulative Distribution Function (CDF) representing the known data distribution.  Each point
%estimate  takes into account not only near points but a random
%variable transformed according to the CDF.  
Each image that is created has both the covariance of the known data but
also the variance. 
By rerunning
the procedure an equi-probable  result can be obtained.  Each
estimated model give more realistic fluid flow measurements and by comparing
and contrasting variability can be assessed.
\par
We can achieve the same goal in our geophysical formulation
(\ref{eq:geophysics}) by replacing the $\zero$ with a random vector $\bf v$,
\beqa
\bf d \pox  \bf L  \bf m  \nonumber \\ 
\bf v \pox \epsilon \reg \bf m\label{eq:rand} .
\eeqa
By adjusting the distribution of $\bf v$ we can change the distribution of
$\bf m$. For example, let's return to the SeaBeam example.
Figure~\ref{fig:distrib} shows four different model estimations
using a normal distribution and various values for the variance.
Note how the texture of the model changes significantly, if we look
at a histogram of the various realizations (Figure~\ref{fig:movie.distir}) 
we see that the correct
distribution is somewhere between our second and third realization.
%Using the same normal score transform common in geostatistics we
%could emulate any distribution.

\activeplot{distrib}{width=6in,height=4.5in}{ER}{Four different realizations
with increasing Gaussian noise for $\bf v$ in fitting goals (\ref{eq:rand}).}

\activesideplot{movie.distir}{width=2.0in,height=2.0in}{ER}{Histogram of
the known data (solid line) and the four different realizations of
Figure~\ref{fig:distrib}.}
\par 
Figure \ref{fig:sea.movie} shows eight different realizations with a
random noise level between the second and third example of Figure~\ref{fig:distrib}.
Note how we have done a good job emulating the
distribution of the known data. Each image shows some similar
features but also significant differences (especially note
within the `V' portion of the known data).
\par

\activeplot{sea.movie}{width=6.0in,height=8.0in}{ER}{
Eight different realizations of the SeaBeam interpolation problem and
their histograms. 
Note how the realizations  vary away from the known data points.  }

A potentially attractive feature of setting up the problem in this
manner is that it allows for both space varying covariance functions
(a steering filter or non-stationary PEF) along with a non-stationary
variance. Figure~\ref{fig:non-stat} shows the SeaBeam example
again with the variance increasing from left to right.

\activesideplot{non-stat}{width=2.5in,height=2.5in}{ER}{Realization where
the variance added to the image increases from left to right.}
