In Chapter~\ref{chap:intro} I pointed out that the step size calculated
through a conjugate gradient criteria is incorrect because it doesn't
take into account  ${\bf s_0}$.  To get arround this problem we have
to limit  size and/or regularize the problem more than would be otherwise
needed.  
In addition our ray based 
tomography  problem is sucessiptilbe to large outliers (due
to inaccurate reflector picks) 
and/or bad raypairs.

To test whether these factors have a significant affect on our inversion
I replaced the linear conjugate gradient condition with
the Fletcher-Reeves non-linear
conjugate gradient conditions \cite{opbook} and  a Dennis-Schnabel
line search method \cite{den}.
In addition, 
I replaced the $L_2$ function with a Huber functional \cite{Huber:73}
that is
less sensitive to large outliers \cite{Guitton.sep.100.antoine1}.
The Huber
functional is $L_2$ until some cutoff value and then smoothly switches
to $L_1$ (Figure~\ref{fig:huber}).
The idea is to  compromise between the convergence speed of
$L_2$ and he less sensitive nature of $L_1$  to outliers.

As a test I redid our first non-linear itteration in depth 
and tau using a line search and the Huber functional. As
Figure~\ref{mig.huber.vel1.z} shows our depth result is improved
by using the Huber functional.  Figure~\ref{mig.huber.vel1.tau} shows
the result of using the Huber functional on our tau space 
tomography formulation.
Again we see an improvement over $L_2$ functional.  When I actually
do this I should comment more on the results.