\def\figdir{./Figs}
An important consideration in many geophysical problems, including
tomography, is speed of convergence.
Tomography, even ray based, is computationally intensive so
it is important to minimize the number of steps it takes
to get to a reasonable solution.
One of the reasons for slow convergence is that our regularization operators,
including steering filters, are small in size.  
A classic solution is to reformulate our regularized inversion problem
into a preconditioned problem in 
terms of some new variable $\pvar$ \cite{opbook} and used
in geophysics by \longcite{harlan-pre}.
The goal is to replace our regularization operator $\reg$ 
with a preconditioning
operator $\bold B$ that smooths long distances with each iteration.
The ideal
choice for $\bold B$ would be $\prec$, because if $\bold B = \prec$,
our regularized fitting goals (\ref{eq:missing}) and (\ref{eq:reg-basic}) 
would be
equivalent to our new preconditioned fitting goals \cite{Fomel.sep.95.sergey1},
\beqa
\data &\approx& \lop \prec \pvar \\ \nonumber 
\zero &\approx& \reg \model = \reg \prec \pvar =  \iop \pvar \label{eq:miss-prec} .
\eeqa
The speed up is due not only to the 
preconditioner spreading information long-distances
with every iteration but also to our regularization
equation now being the identity matrix.  
The identity
matrix is its own inverse reducing the condition number of
the matrix we are trying to invert.
% so the
%conjugate gradient algorithm does not have to spend any
%effort trying to invert this portion of the matrix \cite{gee}.

\subsection{Helix transform}
\input{../steer/helix_act}

With the helix we can now solve the missing data problem that took
us 40 iterations as regularized problem, Figure~\ref{fig:qdome-reg-cont}, in
six  iterations at the same cost per iteration  with the same quality result.

\activesideplot[t]{qdome-prec-cont}{width=3.0in,height=3.0in}{ER}{The 
interpolation result after six iterations using the preconditioned formulation  
(\ref{eq:miss-prec}) of the problem.}

