
Non-linear problems are much more difficult to solve than linear problems.
A method that is often successful is to  make some approximations
that will turn the non-linear problem into a linear problem.
We can make
this conversion by doing
a Taylor expansion (ignoring second and higher order terms) around
our initial guess at the slowness field,
\beqa
\bf t \pox  \tomo{0} \slow{0} + \nabla \tomo{0} \ds  \nonumber \\
\bf t \pox   \bf t_0 + \tomo{0} \ds  \label{eq:linear-approx} \\
\dt \pox \tomo{0} \ds  \nonumber
\eeqa
where
$\slow{0}$ is our initial guess at slowness,
$\tomo{0}$ is the Frechet derivative,
a {\it linear} approximation of $\tomo{nl}$ at
${\bf s_0}$,
$\ds$ is the change in slowness,
$\bf t_0$ are the modeled travel times by applying $\tomo{0}$ to $\slow{0}$,
$\dt$ are the difference between the modeled travel times, $\bf t_0$,
 and
the measured travel times, $\time$.
Reflection seismic tomography problems are too large to do direct matrix
inversion  so iterative methods, such as conjugate gradient,
are used instead.
Once we have  used an iterative method to converge to an acceptable
$\ds$, we update our slowness estimate,
\beq
\bf s_1 = \slow{0} + \ds .
\eeq
We can then re-linearize around this new model ($\bf s_1$), constructing
a new tomography operator $\tomo{1}$ and repeat the process.
\par


Unfortunately, the  linearization does not share many of the properties of
its fully linear equivalent.
First, the step length calculation that is valid
in the linear problem is usually not valid in the non-linear problem.
Instead,  we  must take into  account our linearization
point when calculating the step size.
Second, and more importantly, our problem isn't guaranteed to converge.
To see why, let's return to our simple example.
Our operator is based on our initial slowness estimate, which is in
error. In the case of the ray example, this means that we are back projecting
along our guess at the ray trajectories, not their true trajectories.  
As a result
we will end up  back projecting our slowness changes
to the wrong portions of our model space.
\par
Figure~\ref{fig:first.tomo} shows the velocity estimate after applying
linearized tomography. Generally, we have moved towards the correct
velocity model (Figure~\ref{fig:cor.overlay}) but we can also
see the effects of our  linear approximation. In the upper right portion
of Figure~\ref{fig:first.tomo} the velocity has been incorrectly increased.
If we look at the ray connecting ($z=1.6,x=0$) and ($z=0,x=9.5$) in
Figures~\ref{fig:init.overlay} and \ref{fig:cor.overlay} we can
begin to see why.  Our guess at  the raypath  indicates that
30\% of the ray length is spend  in the lower layer while
in fact 50\% is traveling through higher velocity layer.  
To account for
the traveltime discrepancy the inversion has increased the velocity in the
in the upper portion of the model.


\sideplot{first.tomo}{width=3.0in,height=3.0in}{The tomography result
using the slowness model and raypaths of Figure~\ref{fig:init.overlay}
as our initial estimates.}


A second inconsistency 
is that equation (\ref{eq:linear-approx}) 
is  an example of {\it transmission tomography} rather than {\it
reflection tomography}. Reflection tomography attempts to invert
the slowness field from raypaths that go from a known source, to
a {\it unknown} reflection point in the subsurface, 
to a known receiver (Figure~\ref{fig:ref.tomo}).
Not knowing the reflection point introduces a whole new set
of unknowns into the inversion.
Figure~\ref{fig:intro-rays}  shows a synthetic anticline model.  Overlaid
in solid lines are the correct reflector position and the correct raypaths
to these reflector position.  The dashed lines are the reflector positions
and raypaths that would be estimated using a $s(z)$ initial slowness model.
Note that our guess at the reflector position is in significant error,
making our guess at the raypaths, and the corresponding tomography operator,
in more error than its tomography counterpart. 
In addition, traveltime error can be
accounted for by shifting the reflector, changing the slowness field,
or some combination of the two.  The result is that reflection tomography
is much more likely to get stuck in local minima and maxima than 
its transmission counterpart.

\sideplot{ref.tomo}{height=3.0in,width=3.0in}{A schematic of the reflection tomography problem. We are inverting for slowness from arrival times that
go from a known source $s$ location, 
to an {\it unknown} reflector local $r$, to a known receiver location $g$.}
a known 

\plot{intro-rays}{height=3.0in, width=6.0in}
{The solid lines show raypath through the correct velocity and the correct
reflector position.  The dashed lines are raypaths through
our initial model and our initial reflector positions.  Note that
our estimated raypaths have significant error, therefore our tomography
operator will have significant error.}


%a one variable problem.
%Imagine our non-linear problem has a fairly complicated error surface,
%Figure~\ref{fig:newton}.
%The linearization method can only see in the region around ${\bf s_0}$,
%it uses a parabolic approximation for the rest of the error surface \cite{menke}.
%If we are at point `A' on Figure~\ref{fig:newton} conjugate gradient would tell us that the minimum is at the
%bottom of the dashed curve, not
%the true minimum. If we were at location `B', our parabolic
%approximation would actually lead the solution towards a local maxima,
%and at if
%our starting guess is
%`C'  we would end up near a local minima.
%
%\sideplot[t]{newton}{width=3.0in}{A synthetic error surface for a
%non-linear problem.  The dashed line represents the error surface we approximate
%assuming that we are at `A'.  At `B' our approximation would take us
%away from the true minimum, and at `C' our guess at the appropriate step
%size would be in significant error.}
\par 
The possibility of converging to a local minima is well known.
The general
solution is to try to get to as accurate an initial model as possible 
(to move our initial point as close the global minima as possible).
This is usually done by first applying
conventional velocity analysis methods such as semblance analysis and 
interval velocity estimation by Dix \cite{GEO20-01-00680086} or some 
derivation on Toldi's \shortcite{Toldi.sepphd.43} approach. 
In addition, smaller step sizes are used to
avoid going outside the valid range of our linear approximation.
Both of these approaches have potential negative consequences.
A smaller
step size means more expensive non-linear iterations.
In addition conventional velocity estimation often does not provide
a good initial estimate and might in fact lead us to a 
local maxima.
\par
