\def\figdir{./Figs}
In tomography we linearize around an initial estimate
of the slowness field.   In ray based tomography our linearization assumption
means we have to assume stationary raypaths.
Using this assumption, in 2-D,  we can write
an equation for the traveltime  along a ray as:
\begin{equation}
t = \int_{{\bf L}}  s(z,x)  dl \label{eq:init}  ,\end{equation}
where $s(z,x)$ is our slowness field, $t$ is our measured travel time,
and we integrate over the raypath ($\bf L$).
In discrete space, we will approximate the continuous function
as a series of straight ray segments through constant slowness.
We rewrite equation (\ref{eq:init}) in terms of these ray segments as:
\begin{equation}
\widetilde{dt} = \sqrt{\left(\widetilde{dz}\widetilde{S}\right)^2+
\left(\widetilde{dx}\widetilde{S}\right)^2} , \label{eq:segment}\end{equation}
where
$\widetilde{dt}$ is the travel time of the ray segment,
$\widetilde{S}$ is the slowness at the ray segment location,
$\widetilde{dx}$ and $\widetilde{dz}$ are the change in $x$ and $z$ along 
the segment.
We
can take the derivative with respect to slowness:
\begin{equation}
\frac{ d ( \widetilde{dt}) }{ ds } =  
\sqrt{\widetilde{dz}^2 + 
\widetilde{dx}^2}  {d \widetilde{S} \over d s} \label{eq:back}\end{equation}
and end up with a linear relationship between the change in slowness 
and a change in our traveltime for that ray segment.
We can use (\ref{eq:back}) as the basis
for a linear operator $\tomo{z,ray}$ which relates
changes in the slowness field  $\ds$ to errors in the traveltimes $\dt$,
\begin{equation}
\dt =  \tomo{z,ray} \ds .\end{equation}
If we were doing
a transmission tomography problem,  
such as medical imaging or cross well seismology, we
would  have all  we need to invert for a new slowness model.
Unfortunately, in reflection tomography, there is an added unknown:
the actual point where the ray reflects.
Any  error in our traveltime can be accounted for by either
perturbing the slowness model or by moving the reflector.
One solution to this coupled relationship is
to  reformulate the problem in terms of both
reflector movement ${\bf \Delta r}$ and slowness changes $\ds$
obtaining a new objective function $Q$,
\begin{equation}
Q(\ds,{\Delta r}) = \| \dt - \epsilon_1 \tomo{z,ray} \ds - \epsilon_2 
\bold R{\bold  \Delta r} \|^2 \end{equation}
or more conveniently in terms of the fitting goals
\beqa
\dt \pox \epsilon_1 \tomo{z,ray} \ds \\ \nonumber
\dt \pox \epsilon_2   \bold R {\bold \Delta r} ,
\eeqa
where $\bf R$ is an operator that relates reflector movement
to changes in traveltime.
To find $\bold R$
let's  consider a ray reflecting at angle $\theta$, at `A',
and  a reflector movement of
$dr$ (Figure~\ref{fig:dl}).
Snell's law must obeyed at the reflector.
Therefore reflector movement will be normal to the reflector
at `A'.  Simple geometry
demonstrates that the increase
in the ray length due to $dr$ is $dr  \cos \theta$.
We can convert the change in length to a change in travel time by
multiplying by the local slowness $\sref$. If we account for
both the incident and reflector ray we end up with the linear
relationship for a ray between the change in reflector position $dr$ and
the change in traveltime for the ray $dt$,
\begin{equation}
d t =  2  \sref d r \cos \theta .\end{equation}
\activesideplot{dl}{width=4.0in,height=3.0in}{NR}
{Given the initial ray path (solid lines) the change in
the raypath length is  $cos \theta$  times the
change in the reflector position $dr$.}

Instead of directly inverting for reflector movement and slowness
a commons solution
was to handle the slowness/reflector position
coupling by freezing one component (such as
reflector position) inverting for the other and then reversing the procedure.
This approach has a tendency to be unstable
for large reflector movements. 
  As a result,
\longcite{GEO56.04.04830495} 
and \longcite{Trier.sepphd.66}
proposed  avoiding  inverting
directly for the reflector position by introducing a new operator,
$\bold H$, that maps reflector movement to  $\ds$, and changing the reflector
movement tomography operator to:
\begin{equation}
\tomo{z,ref} =  \bold R \bold H \label{eq:chain} .\end{equation}
As mentioned earlier, the reflector movement is normal to the
reflector at the reflection point.
Therefore, the change in reflector position is going to be
function of how much the slowness field changes along a ray
whose takeoff angle is normal to the reflector (the zero offset ray).
%The change in reflector position due to a change in slowness can be
%approximated by considering a single ray segment along the zero offset ray.
The arrival time at zero offset is independent of velocity.
Therefore we can write an expression between our initial ray
and the ray through our updated medium,
\beqa
l_{\rm new} s_{\rm rms,new} = t &=& l s_{\rm rms} \\ \nonumber
(l + dl ) (s_{\rm rms} + ds_{\rm rms}) &=& l s_{\rm rms},
\eeqa
where $l$ is the length of the zero offset ray through
the initial medium, $s_{\rm rms}$ is the RMS slowness of the media
along the ray, and   $s_{\rm rms,new}$ and $l_{\rm new}$ are the
same two quantities through the updated medium.
If we ignore the second order term we end up with the expression,
\begin{equation}
\frac{dl}{ds_{\rm rms}} = - \frac{l}{s_{rms}} .\end{equation}
If we consider each ray segment independently we end up with
%\widetilde{dl_{\rm new}}\widetilde{S_{\rm new}} &=& t =
%\widetilde{dl} \widetilde{S}  \\ \nonumber
%\left(\widetilde{dl} +\frac{\widetilde{dr}}{ds} \right) 
%\left(\widetilde{S} + \frac{\widetilde{S}}{ds} \right) &=& 
%\widetilde{dl} \widetilde{S}  
%\eeqa
%where $\widetilde{S_{\rm new}}$ is the updated slowness, $\widetilde{dl_{\rm new}}$
%is the updated ray length,and  $\frac{\widetilde{dr}}{ds}$ is the change in 
%reflector position due to the ray segment.
%We can ignore the second order term and end up with
\begin{equation}
\frac{\widetilde{dr}}{ds} \approx - \frac{dl}{\widetilde{S}}  \frac{d \widetilde{S}}{ds} .\end{equation}
Combining the two
portions of our tomography operator (Figure~\ref{fig:schematic})
we get our complete tomography operator relating slowness changes
to traveltime errors,

\begin{equation}
\dt \approx \ztomo \ds \label{eq:tomo-base} .\end{equation}
\activeplot{schematic}{height=3.0in}{NR}{The two portions of the back
projection operator for a given offset, common reflection point (CRP), 
and reflector position.  
$\tomo{ray}$ is the ray-pair from the source $\bf s$ to the
reflection point $\bf r_0$ to the receiver $\bf g$.
$\tomo{ref}$  is the raypath who is normal to the reflector at $\bf r_0$.}

%\par
%A problem with this formulation is that our $\tomo{ref}$ operator
%creates an inconsistency in the construction of $\tomo{ray}$.  To
%show this, let's consider  the reflection portion of the tomography
%operator.  The left portion of Figure~\ref{fig:warp-c} shows
%a single ray reflecting in the subsurface and the reflector movement $dr$
%indicated at the reflection point. $\tomo{ref}$ says that we need to warp
%our coordinate system (the right portion of Figure~\ref{fig:warp-c}) so that
%our reflection point is consistent with $dr$.  What we don't account for
%is that by warping our coordinate system, we have also increased the
%length of every ray segment used in $\tomo{ray}$.  In the next section,
%I will introduce a way to do tomography in a coordinate system where our
%reflector movement is much more limited, and therefore this inconsistency
%is significantly reduced.
%
%\plot{warp-c}{height=3.0in,width=6.0in}{The left portion shows
%a ray reflecting in the subsurface and a change in reflector position $dr$
%indicate at the reflection point.  The right panel is the warping
%of the coordinate system implied by $dr$.}

%We can   can be approximated by integrating over the
%image ray (the ray
%perpindicular to reflector at the reflection point). 
%Increasing the slowness, decreases the ray length and
%therefore causes a negative change in reflector position.
%For each ray segment of length $ \widetilde{d l}$
%we can approximate the change 
%in ray length $\widetilde{dr}$ caused by a change in slowness
%as
%\beq
%\frac{\widetilde{d r}}{d s }  = -  \frac{\widetilde{d l}}{\widetilde{S}} \frac{d \widetilde{S}}{ds}  
%\eeq
%where $\widetilde{S}$ is the slowness at the ray segment.
%
%
%
%
