%\section{THEORY}
Following the method described in Clapp \shortcite{Clapp.sep.97.bob2},
we  began by linearizing the tomography problem around an initial
guess at our slowness model $\slow{0}$. We assumed ray stationarity
and described the change in travel time ($\dt$) as being linearly
related to our change in slowness ($\ds$):
\beq
\dt \approx \tomo{z} \ds .
\eeq
$\tomo{z}$ is composed of two portions.  The first, $\tomo{z,ray}$  simply
applies
\beq
\delta t  =  \tilde{l} \delta s \label{eq:depth},
\eeq
or that the change in the  travel time is ($\delta t$) is equal to change in 
slowness  ($\delta s$) times length of the ray of the ray segment ($\tilde{l}$)
 of the ray
connecting the source, reflector, and the receiver.
The second component, $\tomo{z,ref}$,  can be thought of as a chain of two
operators: the first maps our change in slowness ($\ds$) into reflector 
movement, the second maps the reflector movement into our change in travel 
times ($\dt$) \cite{Trier.sepphd.66}. This second term amounts to performing
residual migration and can be done by back projecting a ray located
at the reflection point perpendicular to the reflector \cite{SEG-1994-1338},
Figure~\ref{fig:schematic}.  Taking both components into account our
tomography fitting goal becomes
\beq
\dt \approx ( \tomo{z,ray} + \tomo{z,ref} ) \ds .
\eeq

\plot{schematic}{height=3.0in}{The two portions of the back
projection operator.  $\tomo{z,ray}$ is the pair of rays through
$\slow{0}$ from the source to the
receiver that obey Snell's law at the reflector. $\tomo{z,ref}$  is the raypath
from this reflection point to the surface.}

%\subsection{Smoothing slowness rather than change of velocity}
In general, the tomography problem is under-determined and requires
some type of regularization.  Ideally, this regularization should
be the inverse model covariance \cite{tarantola-ipt} but that is not
readily available.  In many cases we do have well logs, initial
migration surfaces, or a geologist's model of the region that can
at least indicate the trend that velocity should follow.  Following
the method described in Clapp, et al. \shortcite{Clapp.sep.100.bob3} we
can build a space-varying operator composed of small plane wave annihilation
filters that can smooth our velocity along this predetermined trend.

The problem is that our model is not slowness, but change in slowness.
To a degree, we can get around this problem by following the method 
similar to the one described by
Bevc \shortcite{Bevc.sep.82.145}.  
We start by stating our goal to smooth the slowness
field:
\beq
\zero \approx \reg \slow{} 
\eeq
where $\reg$ is our steering filter operator.  But $\slow{}$ is actually
$\slow{0} + \ds$, so we can write a new regularization goal as
\beqa
\zero \pox \reg (\slow{0} + \ds) \\ \nonumber
- \reg \slow{0} \pox \reg  \ds  .
\eeqa

A problem with this method is 
where the adjoint of our modeling operator ($\tomo{z}^{'}$) 
 does not contribute at all to the model we
can introduce artifacts.  Our best solution to date for
this problem is to introduce a smooth masking operator that tapers
off to zero in locations unaffected by $\tomo{z}^{'}$.

%\subsection{Preconditioning}
The proposed regularized tomography problem still has the problem
of slow convergence.  By reformulating the problem in helix space
\cite{helix}, we can take advantage of 1-D theory to change our
regularized problem into a preconditioned one.  We start by defining
a new variable $\pvar$: 
\beq
\pvar = \reg \ds .
\eeq

By applying polynomial division to our steering filters, we can
create $\prec$ which becomes a smoothing operator.
We can then rewrite our fitting goals as 
\beqa
\dt \pox (\tomo{z,ray} + \tomo{z,ref}) \prec \pvar \nonumber \\
- \reg \slow{0} \pox \epsilon \pvar  \label{eq:depthtomo}.
\eeqa


%\subsection{Tau tomography}
In  depth tomography we must constantly deal with the depth-velocity
ambiguity problem.  Put another way, we are simultaneously 
trying to estimate both a focusing  ($S_f$) and a mapping ($S_m$)
slowness.  Biondi et al.\shortcite{Biondi.sep.95.biondo1} showed
that by mapping ($z,x$) into ($\tau,x$) through
\beq
\tau(z,x) = \int_0^z 2 S(z',x) \delta z'
\eeq
we can write  a {\bf focusing eikonal} equation which only
indirectly depends on the mapping velocity
\beq
4\left( {\partial t (\tau,x) \over \partial \tau }\right) + S_f(\tau,x)^{-2} \left({\partial t (\tau,x)
\over \partial x} + \sigma_m(\tau,x) {\partial t (\tau,x) \over \partial \tau}\right)^2 =1
\eeq
where $\sigma_m$ is the differential mapping operator defined as
\beq
\sigma_m(\tau,x) = \int_0^\tau S^{-1}_m (\tau',x) {\partial \over \partial x}S_f(\tau',x) \delta \tau' .
\eeq
From this eikonal equation we can derive a new  relation for the
change in  travel time  due to a change in the focusing velocity:
\beq
\delta t = \left(  {\tilde{\delta x}^2 \over \tilde{l}}  \right) \delta s -
\left(\tilde{\delta x} ( \tilde{\delta \tau} - \tilde{\delta x}  
\tilde{\sigma_0}(x,\tau)) \right) \delta \sigma \label{eq:tau}
\eeq
where
%?? YOU ARE DOING SOMETHING WEIRD HERE
$\tilde{\delta x}$ and $\tilde{\delta \tau}$ are the change in 
$x$  and $\tau$ 
position of the ray segment,
$\tilde{\sigma_0}(x,\tau)$ is the differential
mapping factor of our initial
slowness model at the ray location, and
$\delta \sigma$ is defined as
\beq
\delta \sigma(\tau,x) = {\sigma_0(\tau,x) \over s_0(\tau,x)}  
\delta s(\tau,x) - 2 s_0(\tau,x) {\delta (\delta z(\tau,z)) \over \delta x} 
\eeq
where 
\beq
\delta z(\tau,z) = \int_0^\tau{ \delta s(\tau',x) \over 2 s_0^2(\tau',x)} \delta \tau .
\eeq

We now have a way to back project travel time errors and can
write a new set of fitting goals,
\beqa
\dt \pox (\tomo{\tau,ray} + \tomo{\tau,ref})  \prec \pvar \nonumber \\
- \reg \slow{0} \pox \epsilon \pvar  \label{eq:tautomo}
\eeqa
where $\tomo{\tau,ray}$ and $\tomo{\tau,ref}$ use (\ref{eq:tau}) rather
than (\ref{eq:depth}) to back project.

%Problem is method is that it is hard to distinguish
%between a velocity error and reflector mispositioning.  One solution
%is to put off reflector positioning term at concentrate on
%fucousing.  As Clapp \shortcite{Clapp.sep.100.bob3} shows this makes
%our linearization more stable from iteration to iteration.
%To do this we do stuff:
%
%Give the eikonal equations for tau
%
%
%We get a new set of equations:
