For the first comparison I applied a single non-linear iteration of
depth, fitting goal (\ref{eq:tomo-base}), and tau,
fitting goal (\ref{eq:tomo-base-tau}),  tomography.
Figure~\ref{fig:iter1-comp} shows  $\ds$ and $\bold v$ for both.
The $\ds$ models are quite similar.  The depth case has changed
the slowness in the lower portion of the anticline less, which is
more accurate.  The tau case has done a better job finding the edges
of the anticline. In addition the tau velocity model has an interesting
artifact (the low velocity layer curving down) due to introducing the slowness
change in tau rather than depth and then mapping back.




%Generally there is a tradeoff between model smoothness and fitting
%our moveouts (which will be covered in more detail in Chapter~\ref{chap:reg}).
%The depth $\ds$ result has a significantly higher spatial frequency, but 
%even allowing a higher frequency $\ds$ model I was not able to match
%moveout errors as well as in the corresponding tau inversion.  The
%most important difference to observe between these two results is how
%far our $\ds$ changes are propagated.  In the depth case I have
%changed the near surface slowness model significantly away from the anticline,
%while in the tau image our changes have been much more localized to the
%anticline structure itself.
%
\plot{iter1-comp}{height=6.0in, width=6.0in} {Velocity change (top) and 
velocity (bottom) after
one iteration of depth (left) and tau (right) tomography.}

\par 
If I then migrate with the
two velocity models we get very similar results.
Both results Figures~\ref{fig:res.vel1.deltaz} and \ref{fig:res.vel1.deltatau}
show significantly better placement of the reflectors than the initial
migration result, Figure~\ref{fig:res.vel0}.
The tau result, has done a better job positioning the reflector reflectors and 
shows less movement in the CRP gathers than its depth counterpart.
After a single iteration it is difficult to make a conclusive statement
on which methods is better.  

%In the case of depth (Figure~\ref{fig:res.vel1.z}) 
%the higher frequency component of our slowness
%field has caused our migrated image quality to  deteriorate. 
%The moveout in the CRP gathers within the anticline has been reduced but
%at CRP gathers away from the anticline I have created moveout errors. 
%The tau migration result is significantly better
%(Figure~\ref{fig:res.vel1.tau}).  We have created
%some minimal additional moveout in the CRP gathers away from the anticline%,
%but much less than the depth counterpart.  In addition, our overall image
%quality is better, the positioning of the reflectors is more accurate, and
%I reduced the residual moveout in our gathers more than the depth counterpart.

\plot{res.vel1.deltaz}{height=3.0in, width=6.0in} {Migration result
after one iteration of depth tomography. We have reduced the moveout
and positioned our reflectors within the anticline significantly better
than in Figure~\ref{fig:res.vel0}.}
 
\plot{res.vel1.deltatau}{height=3.0in, width=6.0in} {Migration result
after one iteration of tau tomography. We have reduced the moveout and
positioning errors compared to Figure~\ref{fig:res.vel0}.  Positioning
of all but the basement reflector is better than its depth counterpart,
Figure~\ref{fig:res.vel1.deltaz}.}
