%\section{TOMOGRAPHY EXAMPLE}
In general,
tomography problems are not as straightforward as the problems presented
above.  First and foremost, the tomography problem is non-linear. 
Perturbations in the slowness model change the raypaths, 
therefore the mapping opperator.
Generally, we get around this problem with a two step procedure.
We first assume stationary raypaths
and use a linearized back projection operator.  We then add an outer,
non-linear raytracing loop to the problem\cite{GEO56.04.04830495}. This
procedure often works, but can easily diverge if our initial guess at
the velocity model (therefore our raypaths) are to far from the correct model.
In addition, we must deal
with the inherent velocity-depth coupling problem: any changes
in traveltimes can be caused by either reflector movement or slowness
model changes.  
As a result we must take into account reflector movements
when evaluating the linearized tomographic operator \cite{Trier.sep.66}.
Following the methodology of Stork and Clayton \shortcite{GEO56.04.04830495}
we cast our tomography problem in the form:
\begin{equation}
\bold{\Delta t_m^i } \approx  \bold{L_s^i\Delta s} + \bold{G^i\Delta R^i} =
(\bold{L_s^i} + \bold{G^iH^i})\bold{\Delta s},
\end{equation}
where
\begin{description}
\item [$\bold{\Delta t_m^i }$] is the difference between the modeled
and the correct travel times,
\item [$\bold{L_s^i}$] is the back-projection operator along our modeled
ray paths,
\item [$\bold{G^i}$] maps changes in reflector position to changes in traveltimes,
\item[$\bold{\Delta R^i}$] is the change in reflector position,
\item [$\bold{H^i}$] maps slowness changes to reflector movement, 
\item[$\bold{\Delta s}$] is the change in slowness.
\end{description}
We handle the reflector movement component by calculating and back-projecting
a zero-offset ray located at the reflection point (Figure~\ref{fig:zero}).


Finally, we  add in our preconditioning operator
to obtain our final set of tomography goals,
\begin{eqnarray}
\bold{\Delta t_m^i} &\approx& (\bold{L_s^i} + \bold{G^iH^i})  \bold{A^{-1}} \bold{p} \nonumber \\
\bold{0} &\approx&  \epsilon \bold{Ip} \label{eqn:tomo.depth} .
\end{eqnarray}

%\subsection{Synthetic Example}

To test the steering methodology versus a more tradition isotropic
smoothing approach  we  created a synthetic  anti-cline 
velocity model with six
reflectors (Figure \ref{fig:depth-overlay-vel-cor}).  To simulate 
picked traveltimes  we shot rays through the correct
model using a
paraxial ray tracing code \cite{SEG.1994.1406} 
and found ray pairs where Snell's Law was obeyed
at the reflectors (\ref{fig:rays}).

\sideplot[t]{depth-overlay-vel-cor}{width=3.5in}{Correct velocity
model overlain by simulated reflectors.}



%MODEL
%\sideplot[t]{overlay-vel-cor}{width=3.5in}{Initial model with reflectors superimposed.}
%\sideplot[]{rays-vel0}{width=3.5in}{Starting model and reflector position with 
%model rays superimposed.}



We use as our starting guess a  $v(z)$ velocity
model calculated by averaging over $x$ at each depth. We  map migrated the
correct depth surfaces according to our initial guess at the velocity model
to obtain our first guess at reflection position
(Figure \ref{fig:depth-overlay-vel0}).
We then calculated the dip of each map migrated reflector and created a 
velocity dip field for the model, Figure~\ref{fig:angles}. This dip field
is then used to construct our steering filter operator.
For our travel-time errors we 
e shot and matched rays using the initial model and 
map-migrated reflector position.
The difference between the modeled and correct reflection times 
(Figure \ref{fig:time-diff}) is used as the input to 
our tomography problem  fitting goals (\ref{eqn:tomo.depth}).

\sideplot[t]{depth-overlay-vel0}{width=3.5in}{Initial velocity
model and the corresponding map migrated reflector position.}

\plot{time-diff}{width=5.0in} {Travel time errors using initial guess at
velocity and reflector position. Top reflector, top-left; Second reflector,
top-center, etc.}

Figure~\ref{fig:one-to-one} shows the result of one non-linear
itteraton using both the steering filters and an inverse laplacian operator
as a preconditioner.  Note how the inverse laplacian result gives too high
and spreads latterally too far velocity for the anti-cline. Next look at 
the map migrated reflector positions using the new velocity model.
The steering filter reflector positions are closer to the correct reflector
position, especially at that edges and at at the center of the anomaly.

If we perform two more non-linear itterations the results are similar 
(Figure~\ref{fig:tomo-compare}). The steering filter preconditioned
result has done a better job in delineating the  edges of the
anti-cline, getting
the correct velocity structure within the anti-cline, and in correctly
positioning the reflectors.


\plot{tomo-compare}{width=5.0in}{The result of 4 non-linear itterations of
tomography.  The right panel uses steering filters as a preconditioner,
the left panel uses a issotropic smoother for preconditioning.}
