%\section{Theory}
%We can write a fundamental relationship between a slowness field and
%the travel times we would collect at some known receiver positions.
%Given  source and receiver locations we can imagine constructing an
%non-linear operator, $\tomo{n}$,
%that  can adequately describe transmission through a slowness
%field and our travel times
% inversion 
There are two problems with this approach.  First, Newton's method
is only guaranteed to converge to a local minima, we hope that by
applying regularization \cite{Clapp.sep.100.bob1} we can avoid this problem.
And second, we are only using the first term in our Taylor expansion, which
means that when our higher order derivatives are large, are descent
direction will be wrong, and we will converge at a much slower rate.
When using rays,  
this problem occurs when the initial guess at ray paths and
reflector locations are too far from their {\it correct} locations.
\par
We can obtain a measure of how inaccurate our linear approximation is
by looking at how much our linearized tomography operator changes
from non-linear iteration to non-linear iteration (the difference 
erence between $\tomo{0}'$ and $\tomo{1}'$.) The smaller the difference,
the more accurate our linearization, and
the less likely our estimate well diverge.

By forming our tomography in ($\tau,x$) rather than ($z,x$) space,
we reduce  the change in $\tomo{1}'$ from $\tomo{0}'$.  The fundamental
reason is that our data is in time rather than depth.
In depth,    reflector positions and layer boundaries 
change significantly from 
iteration to iteration, while in tau, they hardly 
change at all \cite{Biondi.sep.95.biondo1}.
%n addition, even though we are attempting to get an image in depth,
%are data is in time. From non-linear
%iteration to iteration changes in velocity at $z_a$, forces us to change
%the location of structure at $z >z_a$.  For horizontal layering a change
%at $\tau_a$ does not affect the location of $tau >$tau_a$. When we introduce
%dips