For constructing the steering filters I used the same methodology as 
described in Chapter~\ref{chap:reg}.   I first calculated the dip along 
the nine, non-salt reflectors
and then interpolated to the entire model space.
Figure~\ref{fig:amp-vel0} shows the interpolated dip field and
the result of applying $\frac{1}{\bf A \bf A'}$ 
to random noise.  As you
can see the preconditioning operator  tends to create low frequency
changes fairly flat in the left portion of the model and more `U' shaped
changes to the right of the salt.

\plot{amp-vel0} {width=6.0in,height=3.0in}{The left panel is the dip field
used for the first iteration of tomography. The right panel shows the
result of applying $\frac{1}{ \bf A \bf A'}$ 
to random noise.}


\par
The salt and salt boundary added additional complications.  
The sediment  velocity abutting the salt dome  is
significantly different than the salt velocity. Using fitting
goals (\ref{eq:almost}) would create a smooth transition
from the sediment layers to the salt velocity instead of the sharp
contrast that often occurs when salt intrudes into
sediments of significantly lower velocity.  As a result it was necessary to 
make a slight modification to the tomography fitting goals.
\par
I introduced a new diagonal operator $\bf V$,  which is large
where we want model smoothness and becomes smaller as we approach
the salt boundary (Figure~\ref{fig:v-amp}).  
Adding this operator to our model styling goal and then applying
the same preconditioning trick we end up with
\beqa
\dt \pox  \tautomo \prec {\bf V^{-1}} \pvar \nonumber  \\
- \epsilon \reg {\bf s_0}  \pox \epsilon \iop \pvar . \label{eq:tomo2d} 
\eeqa
      
\plot{v-amp}{width=3.0in,height=3.0in}{Weights used for  ${\bf V}$.}

The salt itself was an additional complication. Normally
salt velocity is fairly constant.  Reflector continuity disappears (and
therefore can not be used reliably by the tomography operator) as
we get close to the salt boundary and in lower portions of the model.
Very few  of the raypaths used in tomography pass through the 
salt structure.  As a result the model fitting goal would be dominated
by the model styling goal and instead of a constant velocity function we
would get unrealistic smooth variation in the salt velocity.  To counter
this problem, I followed the common practice of not allowing
the  salt velocity to vary.
