Seismic tomography is a non-linear problem.  
      A standard technique for dealing with this non-linearity
is to iteratively assume a linear relation between
the change in slowness and the change in travel 
times \cite{Biondi.sepphd.64,Etgen.sepphd.68} and then  re-linearize
around the new model.
In ray-based methods, this is 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, the linearization does not cope  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 (tau) coordinate 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
tau back projection operator
is less sensitive to our initial velocity estimate than its
depth counterpart. 
%Therefore, our
%back projection operator changes less from one non-linear to the next,
%making the global estimate less likely to get stuck in local minima.
I finish the chapter by applying and
comparing  tau-space and depth-space tomography
to a simple anticline model.
