\par
Velocity estimation is generally under-determined.
To obtain a pleasing result we impose some type of regularization
criteria  such as
preconditioning\cite{GEO56.04.04830495}, limiting solutions to large
singular values \cite{rowbatham}, or characterizing the model through
a small number of spline coefficients \cite{jun}.
These methods all create velocity models that can correctly model
the recorded travel times, but are often unrealistic when applying
criteria such as geologic feasibility.
\par
To create more geologically feasible velocity models  and to speed
up convergence,
\longcite{GEO56.05.06350653} suggested using varying sized
grid cells.  Unfortunately, such a parameterization is prone to error
when the wrong size blocks are chosen \cite{error}. Other authors have
suggested locally clustering grid cells \cite{GEO56.09.13951404} or
characterizing the velocity model as a series of layers
\cite{GEO61.05.15111523}.
These methods are also
susceptible to errors when the wrong parameterization is chosen.
\par
An attractive alternative
approach is to add an additional model regularization term to our
objective function \cite{Trier.sepphd.66}.   In theory, this regularization
term should be the inverse model covariance matrix \cite{tarantola-ipt}
obtained from some {\it a priori} information sources.
For tomography,
a geologist's structural model of the area, well log information,
or preliminary stack or migration results all could provide useful information.
Incorporating
these varied information sources into our objective function is 
problematic.
%We need to cite the French guy and the tomography paper.
%into the objective.
Geostatisticians have successfully
combined these mixed types of information
\cite{TLE16-03-02530260}.
Unfortunately, the geostatistical approach does not
easily fit within a standard global tomography problem.
More promising approaches were presented by \longcite{error} and
\longcite{struct}.
\longcite{error} incorporated into the objective function a term
encouraging the velocity gradient to follow reflector position.
\longcite{struct} suggested  using {\it a priori} structural information
to create conditional covariance matrices.
\par
%
In this chapter I  take a different approach to adding geological dip information
to velocity estimation.
I start from the same basic assumption as \longcite{error}; that velocity
follows structural dip or some other known trend.
Rather than minimizing the velocity gradient directly
in the objective function I note that
I can approximate  a single dip by
creating small  plane wave annihilation filter  \cite{pvi}.
By adjusting the shape and coefficients within this filter (which
I refer to as a {\it dip penalty filter}),
I can approximate a wide range of covariance responses.
By building a space varying regularization operator
out of these filters (a {\it steering filter}) I can  approximate
a model covariance which is space-variant.
To speed up convergence I reformulate our regularization
problem into a preconditioned problem \cite{gee} using the helix
transform and polynomial division \cite{helix}.

%By replacing these
%symmetric operators with operators that tend to spread information along
%structural dips I can generate more geologically reasonable velocity models.
%We can find the inverse of  these space-varying, anisotropic operators by 
%forming them in helix 1-D space and performing polynomial division.
%These inverse operators can be used as a preconditioner to a standard
%tomography problem, significantly improving convergence speed compared
%to the typical regularized inversion problem.
\par
I will begin by introducing a simple  missing data problem.
I show how the geophysicist and geostatistician find different ways to 
characterize and incorporate the model covariance function into
an inverse problem. One 
geophysics approach, applying a Prediction Error Filter (PEF),
can often obtain a better image.
The downside of the PEF approach is it requires having a field with the
same statistical properties as the 
variable we are attempting to estimate.
I then introduce another way to characterize the model
covariance function, dip penalty filters, that combines the geostatistician's ability to
use disparate and irregular information sources
and the geophysicist's ability to fit into
complex inversion problems. 
I go on to describe how to
find dip penalty filter directions and how to build a steering filter.
I show how 
to speed up convergence by reformulating problems in terms of
preconditioning rather than regularization.  
I conclude the chapter by showing that
a steering filter improves our tomography estimate
using the synthetic introduced in Chapter~\ref{chap:tau}.
