\def\figdir{./Figs}
%\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 
%Solving the non-linear velocity estimation problem through a Newton
%approximate method has two problems.
%First, Newton's method
%is only guaranteed to converge to a local minima, we hope that by
%applying regularization, Chapter~\ref{chap:reg} we can ensure
%that we we converge to the global minima.
%And second, we are only using the first term in the 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 can occur when the initial guess at ray paths and
%reflector locations are too far from their {\it correct} locations
%(cite something).
\par
The accuracy of our linearization (\ref{eq:linear-approx}) 
is the
major factor determining whether our tomography problem will
converge and how many non-linear iterations it will take
to get to an acceptable solution.
For ray based tomography this is a function of how close
our initial guess at our raypaths is to their true locations.
The smaller the difference,
the more accurate our linearization, and
the less likely our estimate will diverge.

By forming our tomography in $\tau,x'$ rather than $z,x$ space,
$\tomo{0}$ is closer to $\tomo{nl}$.
The fundamental
reason is that our data are in time rather than depth.
For ray-based tomography our back projection operator inherently
relies on our estimates of reflector positions.
In depth space,  reflector positions
change significantly with velocity changes.  As shown earlier 
(Figure~\ref{fig:tau-initial})  changing
the velocity the same amount will have very little effect in tau space.
We can extend this argument to our back projection operator.  In the depth 
space
our reflectors are mispositioned and as a result we are back projecting
our errors into these mispositioned locations. In tau, the reflectors
are much closer to the correct position, and the back projection will
be introducing slowness change in more reasonable locations.
%As a result are raypaths our more independent of are initial
%slowness estimate.
%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 

\subsection{Simple test}
To demonstrate how the  tau back projection operator is less affected by
our initial slowness model, I 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.
I assume that I have correctly resolved the lower layer.
For this simple 1-D synthetic that means that the vertical
travel-time to the top and bottom of the layer boundary are
correct.
In depth, because we have not resolved the upper layer,
we will misplace the location 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:ray.vel0}).
I then built the tomography operator for both
tau  ${\bf T_{0,\tau}}$ and depth ${\bf T_{0,z}}$, 
Figure~\ref{fig:operator0}.

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

\activesideplot{ray.vel0}{height=2.0in,width=4.0in}{ER}{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, I ray traced through the `correct' velocity
model in both spaces (Figure~\ref{fig:ray.vel1}) and calculated the corresponding
operators (Figure~\ref{fig:operator1}). 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
has an underlying assumption that we are smoothly converging to
the correct operator.  In  tau space, this assumption is more valid.
With a more complicated model the positioning of our layer boundaries
will be subject to more change, making the tau compared to depth
difference even more dramatic.

      
\activesideplot{operator0}{height=2.0in,width=4.0in}{ER}{The operator calculated from our initial guess
at velocity and the resulting ray paths in depth (left) and tau (right).}

\activesideplot{ray.vel1}{height=2.0in,width=4.0in}{ER}{``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.}

\activesideplot{operator1}{height=2.0in,width=4.0in}{ER}{The operator calculated from the ``correct''
velocity and the resulting ray paths in depth (left) and tau (right).}

\activesideplot{diff}{height=2.0in,width=4.0in}{ER}{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.}