\mhead{VELOCITY SELECTION TESTS}
To test the methodology I applied it to several different velocity models.
The first model is a 2-D synthetic, provided by BP, attempting to emulate 
North Sea geology (Figure~\ref{fig:amoco-vel}).  The model is composed
of several constant velocity layers. For this test I used the entire
velocity model as an input, even though when doing migration each depth
step will by analyzed independently.
Figure~\ref{fig:amoco-hist-overlay} shows a histogram of the velocity
function overlayed by the initial reference velocities (0) 
and those after five (5) and twenty iterations (*).  Note how all of the
major velocities are identified by the method.

\sideplot{amoco-vel}{width=2.0in}{A 2-D synthetic velocity model made up
of constant velocity layers.}
\sideplot{amoco-hist-overlay}{width=2.5in}{A histogram of the velocity
function overlayed by the initial reference velocities (0) 
and those after five (5) and twenty iterations (*).}

For the second example I smooth the velocity in Figure~\ref{fig:amoco-vel}
to obtain the velocity in Figure~\ref{fig:amsm-vel}.  
Figure~\ref{fig:amsm-hist-overlay} shows the selected velocities. 
Note how the initial velocities were clustered around  $3.6 km/s$.
The final velocities are generally more spread out, closer in velocity ranges
with high count.  Also note that in each region the velocity that is most
common is chosen as a reference.


\sideplot{amsm-vel}{width=3.0in}{A smoothed version of the model in Figure~\ref{fig:amoco-vel}.}
\sideplot{amsm-hist-overlay}{width=2.5in}{A histogram of the velocity
function in Figure~\ref{fig:amsm-vel} overlayed by the initial reference velocities (0) 
and those after five (5) and twenty iterations (*).}

To test the method's ability on functions with bimodal distributions
(such as regions with salt), I used the upper (Figure~\ref{fig:zig1-vel})
and lower (Figure~\ref{fig:zig2-vel}) portion of the SMAART JV Sigsbee synthetic.
Figures~\ref{fig:zig1-hist-overlay} and \ref{fig:zig2-hist-overlay}
show the selected velocities. Note how in each example 
the method determined that fewer reference velocities were needed.
In each case a single reference velocity
between the two modes of the distribution (this reference velocity was
kept because some velocities still fall in this region).


\sideplot{zig1-vel}{width=2.0in}{The top portion of the Sigsbee synthetic.}
\sideplot{zig1-hist-overlay}{width=2.5in}{A histogram of the velocity
function in Figure~\ref{fig:zig1-vel}
 overlayed by the initial reference velocities (0) 
and those after five (5) and twenty iterations (*).}


\sideplot{zig2-vel}{width=2.0in}{The bottom portion of the Sigsbee synthetic.}
\sideplot{zig2-hist-overlay}{width=2.5in}{A histogram of the velocity
function shown in Figure~\ref{fig:zig2-vel}
overlayed by the initial reference velocities (0) 
and those after five (5) and twenty iterations (*).}
