Finally, we need to set up our tomography fitting goals in a preconditioned
form.
If we follow the same procedure as we did  for the missing  data problem,
we would first add our regularization goal to our tomography problem:
\beqa
\dt \pox  \tautomo \ds \nonumber  \\
\zero \pox \epsilon \reg \ds ,
\eeqa
where ($\reg$) our steering filter operator.

However, these
fitting goals don't accurately describe what
we really want.   Our steering filters are based on our
desired slowness rather than change of slowness. With this
fact in mind, we can rewrite our model styling goal as:
\beqa
\zero \pox \reg \left( {\bf s} + \ds \right)  \nonumber \\
-\epsilon \reg {\bf s_0} \pox \epsilon \reg \ds .
\eeqa
Our model styling  goal can not be strictly defined as regularization
but we can still do the same preconditioning substitution:
\beqa
\dt \pox  \tautomo \prec \pvar \nonumber  \\
- \epsilon \reg {\bf s_0}  \pox \epsilon \iop \pvar \label{eq:almost} .
\eeqa
%Unfortunately, this set of equations has a problem: it has a tendency
%to blow up in regions outside our ray coverage space. This is
%related to the $- \epsilon \reg {\bf s_0}$  in the
%second fitting goal of  (\ref{eq:almost}).
%When applying 
%$\bf I'$  the regularizer  puts energy into model space points outside
%our ray coverage, or in other words, in the null space of
%our tomography operator.
%The end result is that the conjugate gradient algorithm will end
%up creating $\ds$ that when added to $\bf s_0$ will have
%constant velocity outside ray coverage and will influence
%events inside the ray coverage because of $\prec$ (Figure~\ref{fig:bad}).
%To understand why think of $\reg$ as a vertical
%gradient operator and $\bf s_0$
%as a simple $s(z)$ medium monotonically increasing. When
%we apply $\reg$ to $\bf s_0$ we will get a constant function.
%The second fitting goal of (\ref{eq:almost}) will put in $\pvar$
%something of equal and opposite sign.  We will then converge
%to a solution and apply $\prec$ to $\pvar$ to get $\ds$.
%Unfortunately our $\ds$ will now be a function monotonically decreasing
%at equal and opposite rate of our initial $\bf s_0$ function.  When
%we sum these two to get our final answer we will end up a constant
%slowness model outside our ray coverage region.
%\sideplot{bad}{width=3.0in,height=3.0in}{The velocity when applying 
%(\ref{eq:almost}).  Note that the slowness outside the ray coverage
%is almost uniformly 2.6 km/s and that the $v(z)$ trend has been
%removed.}

%My solution to this problem was to 
%add an additional fitting goal that penalized putting
%energy into the model outside the ray covered area,
%\beqa
%\dt \pox  \tautomo \prec \pvar \nonumber  \\
%- \epsilon \reg {\bf s}  \pox \epsilon \iop \pvar \nonumber \\
%\zero   \pox \epsilon_2 \bold e  \pvar  .
%\eeqa
%
%Where $\bold e$ is  large where we don't have ray coverage and small
%where we have the most ray coverage (this
%very similar to the model weighting operator of Bakus-Gilbert method).  
%To calculate $\bold e$ I first
%first found the hit count at every model location ($\bold h$).  I then
%calculated $\bold e$ using
%\beq
%\bold e = \bold A^{-1} \left({1 \over   a_0 + \bold h} \right)
%\eeq
%%where $a_0$ is a constant.
