\def\figdir{./Figs}
In general the operator $\bf L$ in fitting goals (\ref{eq:rand}) is much more
complex than the simple masking operator used in the missing
data problem.  One of the most attractive potential uses for a 
range of equi-probable models is in velocity estimation.  
As a result I decided to next test the methodology on one of the simplest
velocity estimation operators, the Dix equation \cite{GEO20.01.00680086}.
\par 
Following the methodology of \longcite{Clapp.sep.97.bob1},  I
start from a CMP gather $q(t,i)$ moveout corrected with velocity $v$.
A good starting guess for our RMS velocity function
is the maximum ``instantaneous stack energy'',
\begin{equation}
{\rm stack(t,v)} = \sum_{i=0}^n q(t,i)  .
\end{equation}

We don't wish to weight all times equivalently (some times have reflectors,
some times don't), so we add an additional diagonal weighting matrix, $\bold W$,
found from stack energy, at each selected $v_{\rm rms}(t)$.

Our data fitting goal is to minimize the residual
\begin{equation}
\bold 0
\quad\approx\quad
\bold W
\left[
\bold C\bold u
-
\bold d
\right] .
\end{equation}
We are multiplying our RMS function by $\tau$ so
must make a slight change in our weighting function to give early
times approximately the same priority as later times
we need to multiply are weighting function by the inverse,
\begin{equation}
{\bf W'} =\frac{{\bf W}}{\tau}\end{equation}
Next we need to add in our regularization. I define
a steering filter operator  ${\bf A}$ that tends
the model to introduce velocity changes that follow structural
dip.
I replace
the zero vector with a random vector and  precondition the problem 
\cite{Fomel.sep.95.sergey1}  to get
\beqa
\zero &\approx& {\bold W'} ( \bold C \prec \pvar - \data ) \\ \nonumber
\bf v &\approx& \epsilon \pvar \label{eq:mydix} .
\eeqa
\par
To the methodology I took a 2-D line from a 3-D
North Sea dataset provided by Unocal. 	
Figure~\ref{fig:scale.10.x2} shows four different realizations
with varying degrees of Gaussian random energy put into $\bf v$.
\activeplot{scale.10.x2}{width=6.0in,height=5.5in}{ER}{Four different realization
of fitting goals
(\ref{eq:mydix}) with increasing levels of Gaussian noise in $\bf v$.}

I then chose what I considered a reasonable variability level
and constructed ten equi-probable models.  Note that the general
shapes of the models are very similar.  What we  see are smaller structure
changes.  For examples look at the range between $.7s$ and $1.1s$.
Generally each realization tries to put a high velocity layer in
this region but
thickness and magnitude in the different realizations. 
Figure~\ref{fig:compare-small} shows the two extremes of the realizations
for this window of the data. The  average velocity in the two panels varies by
nearly $100m/s$.

\activeplot{dix-real}{width=6.0in,height=5.5in}{ER}{Four of the ten
different realization
of fitting goals
(\ref{eq:mydix}) with constant Gaussian noise in $\bf v$.}
\activeplot{compare-small}{width=6.0in,height=3.5in}{ER}{The left and center panel
represent to extremes of ten realization applying fitting goals (\ref{eq:mydix}). The right panel is the difference between these two realizations.}
