\mhead{EXAMPLE}
To test the methodology I decided to use a synthetic generated
by BP based on a typical North Sea environment, Figure~\ref{fig:amoco-vel-cor}.
To avoid tomography's  problem with sharp velocity contrasts I
chose to assume an accurate knowledge of the velocity
structure down to $1.8 km$. For the remaining initial velocity structure
I  smoothed the correct velocity.  Figure~\ref{fig:amoco-vel0} shows
the initial velocity model and initial migration.


\plot{amoco-vel-cor}{width=4.8in}{The left panel shows the correct
velocity model. The right panel shows the result of migrating
with the  velocity model in the left panel.}

\plot{amoco-vel0}{width=4.8in}{The left panel shows the initial
velocity model. The right panel shows the result of migrating
with the  velocity model in the left panel.}

I then performed two different series of tomography loops.
In the first case I used a standard approach, without the
constraint on dip of the basement reflector at $4 km$.
Figure~\ref{fig:picked} shows the initial migration with
my pick of the reflector position  overlaid ($\bf r_0$ in
fitting goals (\ref{eq:goals})).  Figures~\ref{fig:amoco-vel1.steer}
and \ref{fig:amoco-vel1.steer-ref} show the velocity and migration
result after a single non-linear iteration of tomography using
both approaches. 
In the first iteration
the velocity structure looks somewhat more accurate without
the dip constraint.
The image tells a different story. Note
how the bottom reflector is much flatter using the dip constraint
condition (Figure~\ref{fig:amoco-vel1.steer-ref}) and the overall image
positioning is a little better. After four iterations, we see a more
dramatic difference. Without the dip constraint condition (Figure~\ref{fig:amoco-vel4.steer})
the velocity model is  having trouble converging, especially along
the right edge. The  bottom reflector is quite discontinuous and
misplaced. The overall image quality is disappointing. With the dip
constraining condition (Figure~\ref{fig:amoco-vel4.steer-dip}
the velocity model is correctly finding the salt boundaries. The 
bottom reflector is fairly flat, consistent, and well  positioned.
The overall image quality is better than the result without
the dip constraint.

\sideplot{picked}{width=3.0in}{The initial migrated model
overlaid by the picked initial reflector position.}

\plot{amoco-vel1.steer}{width=4.8in}{The left panel shows the 
velocity model after one iteration of
`conventional' tomography. The right panel shows the result of migrating
with the  velocity model in the left panel.}

\plot{amoco-vel1.steer-ref}{width=4.8in}{The left panel shows the 
velocity model after one iteration of
tomography with a dip constraint. The right panel shows the result of migrating
with the  velocity model in the left panel.
Note the more continuous nature of the bottom reflector (compared
to Figure~\ref{fig:amoco-vel1.steer}.}

\plot{amoco-vel4.steer}{width=4.8in}{The left panel shows the 
velocity model after four iteration of
`conventional' tomography. The right panel shows the result of migrating
with the  velocity model in the left panel.}

\plot{amoco-vel4.steer-ref}{width=4.8in}{The left panel shows the 
velocity model after four iteration of
tomography with a dip constraint. The right panel shows the result of migrating
with the  velocity model in the left panel.
Note the more continuous nature of the bottom reflector, better constraining
of the salt boundaries, and overall more accurate imaging focusing
and positioning compared to the result without the added constraint (Figure~\ref{fig:amoco-vel4.steer}.}

