To build our steering filter operator we need 
a dip field, the range, and the level of anisotropy 
throughout our model.  In many cases it is important
to have all three of these parameters vary spatially.
For this  simple example we are
going to assign a constant  {\it range} and {\it anisotropy} to the entire
model.
For the dip field estimate we will assume that velocity follows dip and use
the position in which our reflectors image at zero  ray parameter.
Figure~\ref{fig:dir} shows the dip directions based on our migrated
reflector positions using our initial velocity estimate.
%For example, Figure~\ref{fig:dips-z-vel0} shows  the reflector positions
%using our initial velocity and the resulting dip field used
%to construct the steering filter operator.
%\sideplot[t]{directions}{width=3.0in,height=3.0in}{
%Steering filter directions as a function of geologic dip.}

\sideplot{dir}{width=3.0in,height=2.0in}{Steering filter directions as a function of geologic dip. The  bold,
solid lines are estimated reflector positions (taken
from the initial migration), the remaining lines represent dip bars
estimated by interpolating the dips between the reflectors.}





Our initial reflection positions are, of course, in error, so our
dip field estimate is also flawed. Figure~\ref{fig:dips-z}
shows  the initial, correct dip field, and
the error in our estimate. In the tau domain,
where reflector positions are more stationary, our dip
field estimate is better, and  our steering filters (and therefore
the characterization of the model covariance matrix) are more
accurate  (Figure~\ref{fig:tau-dips}). 

%\plot{dips-z-vel0}{width=6.0in,height=3.0in}{Left panel shows the picked
%reflector positions from the initial migration. The right panel
%shows the dip field constructed from the reflectors.}

\plot{dips-z}{width=6.0in,height=2.0in}{Left panel is our initial dip directions. The center panel is the correct
dip field generated from the known reflector positions. The right panel
is the difference between the correct dip field and our estimated
dip field. In each case the dip field is calculated by interpolating between
the reflector above and below the given point. Below the bottom reflector dips
are simply continued.  The large error in the lower portion of the difference 
panel is due to the difference between or estimate, a bowing up of the bottom
reflector, and the flat reflector geometry of the true model.}


\plot{tau-dips}{width=6.0in,height=2.0in}{Left panel is our estimated
dip field in the tau domain. The center panel is the correct dip field
in the tau domain. The right panel is our initial error in our
dip field estimate.  Note that our dip estimate is significantly more
accurate in the tau domain compared to the depth domain 
(Figure~\ref{fig:dips-z}).}
