If our model space is Cartesian mesh or something similar, we face
another problem when doing reflection tomography,
its  large null space.
There are several reasons for this null space:
\begin{description}
\item [shadow zones]  Certain parts of the velocity model simply
might not be illuminated, or illuminated poorly by our acquisition
geometry.  
A classical example is under the edge of a salt body \cite{illumination}.
\item [limited angle coverage]
When the reflection point is unknown, zero angle reflection times
do not contain any information about velocity. As the reflection angle
increases, velocity discrimination becomes easier.
At larger depths, given our limited surface
recording geometry, our angle coverage will decrease (Figure~\ref{fig:ray-limit}).
As a result, larger depth model  components  are often under or unilluminated
\cite{SEG-1999-824}.
\item [not enough equations]
In most reflection tomography methods, we invert a limited set
of positioning errors.  When doing tomography, especially 3-D tomography,
it is possible to have more model points than moveout errors.
\end{description}
\sideplot{ray-limit}{width=3.0in,height=3.0in}
{Given a fixed recording geometry ($s$ being the source
and $g_{\rm max}$ being the farthest
receiver) as we go deeper in depth the angle range that we at
each reflector decreases.}
To deal with
the  null space of the reflection tomography operator we
have two alternative paths
we could follow, reparameterization or regularization.   
\par
Reparameterization  can be done by a coarser sampling of the Cartesian, but
usually involves other parameterizations of the model space.
It
usually takes the form of either a layered model with simple velocity function 
(such as velocity gradient) within the layers \cite{GEO61-05-15111523} 
or reparameterizing the model
in terms 
splines nodes \cite{Biondi.Biondi.sepphd.64}.
The layered model approach is attractive because
it allows a direct link to be kept with an interpreter's
geologic model \cite{GEO61-05-14991510}. It also provides a fairly accurate
description of velocity structure in regions such as the North Sea.
The spline node approach can also be beneficial. It can guarantee
the velocity model will be smooth (necessary for ray based methods)
and with selective placement of nodes can 
allow less freedom in areas where the data provides less information about the 
velocity field.
The downside of both of these approaches is that the parameterization
must be chosen {\it a priori}.  At early iterations our guess
at the velocity model can be in significant error. If we  do 
a poor job of  parameterizing the model we 
could actually hurt, or stop, the problem from converging \cite{error}.
\par
The second option is regularization. 
In the regularization approach we add a second term
to our objective function,
\beq
Q(\ds)=\|\dt-\tomo{0}\ds\|^2 + \epsilon^2\|\reg\ds\|^2 ,
\eeq
where $\reg$ penalizes some definition of roughness in our model.
The regularization approach  has its own drawbacks. The largest
is  that our model space no longer has any direct connection with
our concept of geology. In addition, choosing $\reg$ is problematic and
can introduce unrealistic elements into our slowness estimate.

