%The next step is to see how well our non-stationary filter regularizes
%a tomography problem.
%I constructed a synthetic anticline model
%(Figure~\ref{fig:synth-model})
%with six reflectors,one above the anticline, four within the anticline, and one flat reflector
%representing
%basement rock. For added difficulty, there is a low velocity layer
%between the second and third reflector.
%The model was used to do acoustic wave modeling, with the
%resulting dataset having 32 meter  CMP spacing and   80 offsets spaced
%64 meters apart.If we use as our initial estimation of the
%slowness, an  $s(z)$ function from outside the anticline,
%the reflectors are pulled up
%due to using too low  a velocity within the anticline (Figure~\ref{fig:mig0})%.

%\plot{synth-model}{height=3.3in,width=6.0in}{Left panel is  the synthetic velocity model with six reflectors spanning the anticline.
%The right panel shows the data generated from this  model.}
%
%\plot{mig0}{height=3.0in, width=6.0in}{Left panel is our initial guess
%at the velocity function, the right panel shows the zero offset
%ray parameter reflector
%position using this migration velocity. The correct reflector
%positions are shown as `*'.  Note that reflectors are
%significantly mispositioned.}

A more interesting case is using the non-stationary PEF to regularize
the tomography.
For the test I will use the same anticline model (Figure~\ref{fig:synth-model}) used in 
Chapters~\ref{chap:tau} and \ref{chap:reg}.  
I will start from the same initial velocity model (left panel
of Figure~\ref{fig:mig0}).
and Instead of using a steering filter as $\reg$ in fitting goals 
(\ref{eq:almost})
I will use a non-stationary PEF.



%\sepinput{back}
\subsection{Warping}
\sepinput{warp}

\subsection{Results}

If we  do two non-linear iterations of tomography using
fitting goals (\ref{eq:almost}) each time, we get Figure~\ref{fig:vel3.from0} as
our velocity estimate.  The velocity estimate does a good job
      recovering the anticline shape.  However,
it doesn't do a good job recovering
the low velocity layer.  Migrating with Figure~\ref{fig:vel3.from0} we
get Figure~\ref{fig:res.vel3.from0}. Overlaid on top of the migrated
image are the correct reflector positions.
Overall we did a good job positioning the reflectors.  In addition,
we can see that we have little residual moveout
in our CRP gathers 
(Figure~\ref{fig:res.vel3.from0}) but they are relatively flat.  


\sideplot{vel3.from0}{height=3.0in,width=3.0in}{Velocity model after two
non-linear iterations of non-stationary filter regularization.}

\plot{res.vel3.from0}{height=3.0in,width=6.0in}{Migration result using Figure~\ref{fig:vel3.from0} as the velocity model. The left panel is the zero angle
image overlaid by the correct reflector positions. The right two panels
shows CRP gathers at six, eight, and ten kilometers.}


