In Chapter~\ref{chap:reg}, I introduce a new
way to regularize our tomography estimate
that takes into account our {\it a priori} estimate
of geology.  I start from a simple missing
data problem. 
% I show the geophysicist estimate and use and the
%model covariance matrix to solve the missing data problem.
I compare the conventional way that geophysicists and 
geostatisticians characterize covariance and how they
apply to a missing data problem.
I show how  non-stationarity makes both solutions suboptimal.
%I show how using a non-stationary approach  produces excellent
%results but requires an impractical {\it a priori}  knowledge of the model 
%space.
I then introduce another way to approximate model
covariance, a {\it steering filter}.
I show that a steering filter can accurately handle a non-stationary
model covariance.
%> which has a much more limited knowledge base requirement.
I show how to build a steering filter with as little information
as early migrated reflector positions.
For example, returning to our simply
two layer example (Figure~\ref{fig:cor.overlay}), if we
know the dip of the second layer in the model 
of Figure~\ref{fig:init.overlay} we can do a better job constraining
the slowness field using steering filter regularization, 
Figure~\ref{fig:reg.tomo}.

\sideplot{reg.tomo}{width=3.0in,height=3.0in}{The same tomography
shown in Figure~\ref{fig:first.tomo} now using steering filters
to regularize the velocity estimate.}

\par
Using the steering filter I develop a new tomography objective function.
%for our tomography problem.   
I then compare and contrast the results
of using a steering filter to regularize our tomography
problem versus an isotropic regularizer in both depth and tau.
I show that the most reasonable velocity model,
and the best migrated image, comes from using steering filters
in the tau domain.

