\subsection{Simple test}
To demonstrate how the  tau back projection operator is less affected by
our initial slowness model, we constructed a simple  1-D synthetic.
The model, Figure~\ref{fig:cor}, is composed of two 2.3 km/s zones 
in a constant 2 km/s background.
For this test we assumed that our slowness model  had correctly resolved the
bottom anomaly in vertical travel time.
Our choice of vertical travel time is quite important, as
when doing velocity estimation, we must always
preserve zero-offset travel time.  
In this simple 1-D synthetic, that means that the  vertical travel-time to the
layer boundaries and to the reflector must be kept constant. 
Therefore, in depth, we will misplace the location of
of the bottom high velocity zone but preserve the correct vertical 
travel times to
the layer top and bottom.
After constructing the model we found the 
ray that hit the reflector at 2 km depth, 2 km away from source in
both ($\tau,x$) and ($z,x$) space  (Figure~\ref{fig:vel0}.)
Following the method outlined in Clapp and Biondi
\shortcite{Clapp.sep.97.bob1},
we built the tomography operator for both
tau  (${\bf T^{'}_{0,\tau}}$) and depth (${\bf T^{'}_{0,z}}$), Figure~\ref{fig:operator0}.

\sideplot[t]{cor}{height=2.0in}{Synthetic 1-D velocity function in $\tau$.}

\plot{vel0}{height=3.0in}{Initial guess at the velocity function
overlaid by ray  hitting reflector at 4~km with a half-offset of 2~km. 
Left panel is in depth, right panel is in tau.}

For comparison, we ray traced through the `correct' velocity
model in both spaces (Figure~\ref{fig:vel1}) and calculated the corresponding
operators. By comparing  the correct and initial
operator for  tau and depth, or by looking
at the difference between the two operators (Figure~\ref{fig:diff}), we
can clearly see that our initial guess for our tau operator is overall 
better than our initial guess for our depth operator.  In the upper 
layer, we see marginally more change in the tau operator but at the
lower reflector boundaries (which move in the case of depth but remain
constant in tau) we see significantly more error in depth. 
In addition, the change in reflector position has caused a spike
in the difference panel for the depth case.
Finally, the change in the tau operator is smooth, while the change
in the depth operator shows dramatic jumps.  Our successive relinearizing
have an underlying assumption that we are smoothly converging to
the correct operator.  In  tau space, this assumption seems to be more valid.
With a more complicated model our positioning of layer boundaries, 
will be subject to more change, making the tau compared to depth
difference even more dramatic.


      
\plot{operator0}{height=3.0in}{The operator calculated from our initial guess
at velocity and the resulting ray paths in depth (left) and tau (right).}

\plot{vel1}{height=3.0in}{``Correct''  velocity function
overlaid by ray  hitting reflector at 4~km with a half-offset of 2~km. 
Left Panel is in depth, right panel is in tau.}


\plot{operator1}{height=3.0in}{The operator calculated from the ``correct''
velocity and the resulting ray paths in depth (left) and tau (right).}

\plot{diff}{height=3.0in}{The difference between the operators calculated
from the correct and our initial guess at velocity, for depth (left) and
tau (right.  Note the significant spikes at the reflector and at the lower
layer boundary in the depth case.}

\subsection{Conclusions}
We showed that for this simple model
the tau back projection operator is less affected
by our initial velocity estimate than a depth back projection operator.
We hypothesize that makes tau tomography to some extent {\it more} linear
and, therefore, less likely to diverge.
