Seismic tomography is a non-linear problem.  
A standard technique is to iteratively assume a linear relation between
the change in slowness and the change in travel 
times \cite{Biondi.sepphd.64,Etgen.sep.68.0} and then  re-linearize
around the new model.
In ray-based methods, this amounts to assuming
stationary ray paths and reflection locations to
construct a back projection
operator \cite{GEO56.04.04830495}.  The change in this back
projection operator from non-linear iteration to non-linear
iteration can be thought of as an important second
order effect. 
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Unfortunately, we must still deal with
the coupled relationship between reflector position and
velocity \cite{GEO62-03-09700979,GEO60-01-01640175}.
We can avoid some of the problems caused by this connection
by transforming the problem
into  vertical-traveltime coordinates, tau, space ($\tau$,$x'$).
In the tau domain reflector position is less sensitive to velocity changes.
This modified coordinate
system still allows for complex velocity structures, but significantly
reduces the map migration term in tomography \cite{Biondi.sep.95.biondo1,SEG-1998-1847}.
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In this chapter I begin by deriving a ray based projection operator
in the depth domain. I then perform an analogous derivation  in the
tau domain.
I show that the corresponding
back projection operator
is less sensitive to our initial velocity estimate then its
depth counterpart. Therefore, our
back projection operator changes less from non-linear to non-linear
iteration,  making the global esitmate less likely to get stuck in local minima.
I finish the chapter by showing the results of the tau,
compared to depth, based tomography applied
to a simple synthetic antiicline model.