Now that I have set up our tomography problem it is time to test the
theory and compare the advantage of depth vs. tau tomography. 
The model, Figure~\ref{fig:synth-model},
is a  difficult challenge for tomography, especially ray based
tomography. Our linearization approximation works best
when we are close to the
correct model.  In this case our initial estimate is up
to $1~km/s$ off in places.
In ray tomography we
assume that are raypaths are generally correct.  With this model our
initial  reflector positions are in significant error.
As a result we can't expect to get a satisfactory result after a single
iteration of tomography, 
%and there is a chance that
%we will fall into a local minima and not be able to converge to
%a satisfactory model.
\par
Figure~\ref{fig:res.vel0} is the result of migrating with the $s(z)$ slowness
model (Figure~\ref{fig:mig0}).
The correct reflector positions indicated
with `*'.
As expected away from the anticline our gathers are flat
and we have correctly positioned our reflectors.  Along the edge
of the anticline we see upward curvature in our CRP gathers indicating
that I have used too slow of a velocity at this location.  Below
the center of the anticline, $x=10$, the bottom reflector shows some reverse
moveout, the well known $W$ pattern, better illustrated 
in Figure~\ref{fig:semb-mig0-ref}.

\plot{res.vel0}{height=3.0in, width=6.0in} {Our initial migration result
using a $v(z)$ velocity function from the edge of the anomaly. Right are three
$(z,\theta)$ planes at different x postions.  Note that
the CRP gather at $5.5$ is flat while the other two CRP gathers show 
significant moveout.}

\subsection{First iteration}
\sepinput{first}

%\subsection{Line search}
%\sepinput{huber}

\subsection{Second iteration}
\sepinput{best}

\subsection{Where to go from here?}
Figure~\ref{fig:sem-final} show the semblance 
moveout measure
(the initial semblance is shown in 
Figure~\ref{fig:semb-mig0-ref})  
from the final tau migration, Figure~\ref{fig:res.vel2.deltatau}.
The semblance measures shows that we have  virtually all of the
moveout in the CRP gathers.  The problem is that our updated velocity
model (right panel of Figure~\ref{fig:iter2-comp-vel})  is significantly 
different than  the correct model (left panel of Figure~\ref{fig:synth-model}).
This is often a bone of contention between processors and interpreters.
The velocity model that best focuses the data often is not gelogically
reasonable.
In Chapter~\ref{chap:reg}  I will address how we can reformulate our
tomography problem so we obtain a velocity model that both flattens our CRP
gathers and is geologically reasonable.


\plot{sem-final} {width=6.0in,height=3.6in}{Moveout semblance measured
at the six reflectors on the tau tomography migration result, Figure~\ref{fig:res.vel2.deltatau}.}
