\mhead{SYNTHETIC TESTS}

To test the effectiveness of the method we create a simple anticline
synthetic velocity model, Figure~\ref{fig:model}. We simulated six
reflectors, one on top, four in the anticline, and a basement reflector.
We calculated travel times to all the reflectors for an offset range of
4 kms.  These travel times represent our `recorded travel times'.  For our
initial model we created a v(z) model by taking the lateral average
of the velocity field, Figure~\ref{fig:model}. 

%\plot{model}{width=6.0in}{Left panel is our synthetic model superimposed
%by the six reflectors.  The right panel is our starting guess for our
%velocity function and the map migrated reflector position using this initial
%velocity estimate.}

From this initial model we attempted to invert the velocity function
by three progressively advanced methods:
\begin{description}
	\item [Depth-Standard]: Inverted for a depth model, using an
inverse Laplacian  preconditioner \cite{Claerbout.sep.97.jon2}  for
 $\prec$ in our depth  fitting goals
(\ref{eq:depthtomo})
	\item [Depth-Steering]: Inverting for a depth model, using a
steering filter operator for our preconditioner in  our depth fitting
goals  (\ref{eq:depthtomo})
	\item [Tau-Steering]: Inverting for a tau model, using a
steering filter operator for our preconditioner in
tau fitting goals (\ref{eq:tautomo})
\end{description}


Figure \ref{fig:model1-iter1} shows the result of one non-linear iteration
for all three inversions schemes. All three methods were able to recover
the dome shape after one iteration.  When using a Laplacian smoother
the velocity increase is spread too far both laterally and vertically.
As a result, the bottom reflector is located too deep throughout
the model.  When using steering filters we still have a significant
velocity-depth ambiguity problem, but we have done a little
better job position the bottom reflector.  In the case of tau tomography
with steering filters we have done almost a perfect job after a single
iteration. We have not perfectly recovered the lower portion of the
anticline structure but we have almost completely flattened the bottom
reflector.

\plot{model1-iter1}{width=6.2in}{Left panel is Depth-Standard, middle
is Depth-Steering, right is Tau-Steering.  All after 1 non-linear iteration.}


To see if, and how fast, we could converge to the correct solution
in depth
we performed several more non-linear iterations.  As 
Figure~\ref{fig:model1-best}
shows we did a decent job recovering the anticline with all three methods.
In this case, where the model is fairly simple and we have good travel time
coverage, the big advantage seems to be speed. We got a high quality result
with steering filter tau tomography in a single iteration, while it
took three with steering filter depth tomography, and four when using
the Laplacian and depth tomography.


\plot{model1-best}{width=6.2in}{Top-left: Depth-Standard, after
4 non-linear iterations; top-right: Depth-Steering after 3 iterations;
bottom-left: Tau-steering after 1 iteration; and bottom-right:
a comparison of the reflector positions using all 3 methods.  The solid,
white line is correct reflector position, the small dashes represent
Tau-Steering: large-dashes:Depth-Steering; and the solid black line
is Depth-Standard.}


The smoothness of the anticline was well suited for the Laplacian
so we decided on 
a slightly more difficult challenge  that could better
differentiate between a Laplacian and steering filter regularization.
Our new model
keeps the same basic shape for the model but adds a low velocity
layer within the anticline. 
Figure~\ref{fig:model2} shows the correct, initial, and the result
of  4 iterations using both the Laplacian and steering filters to
precondition the problem.  After 4 iterations the steering
filters have done a much better job recovering the low velocity layer.

\plot{model2}{width=6.2in} {Top, left Our new  model with
a low velocity layer within the anticline; top-right, our starting model;
bottom-left,  Depth-Standard after 4 iterations; bottom-right Depth-Steering
after 4 iterations.}





