The examples shown in Chapter~\ref{chap:reg} were well suited for
using steering filters. Another class of models, one with discontinuities
aren't as well handled by steering filters.  
Figure~\ref{fig:fault-model}
is similar to Figure~\ref{fig:well-logs}, with the exception that we
now have a listric fault in the middle of the anticline structure. If we
follow the same interpolation path as we did in the last section,
the right panel of Figure~\ref{fig:fault-model} is our interpolation result.

\plot{fault-model}{width=6.0in,height=3.0in}{Left panel is a synthetic velocity
model with the beds being offset by a listric fault. The right panel is the
result of interpolating the model using steering filters.}

Instead of reproducing the fault we have created a model
that smoothly changes from horizons on the left  of the fault
to the corresponding horizons to the right side of the fault.  
In many cases, a smooth change is not
only acceptable, but desirable (for example, ray-based methods require
a smooth model).  In other cases the smooth change is unrealistic and
something we want to avoid (salt boundaries and faults which have
significant velocity variation from one side of the fault to the other).
We can avoid some smoothing over the fault by replacing our constant
{\it range} function with one that varies laterally.  For  this example
we can imagine reducing the range of our steering filter as we
approach the fault, Figure~\ref{fig:fault.amp}.  


\sideplot{fault.amp}{width=3.0in,height=3.0in}{Amplitude (range) used to 
construct steering filter for fault model.}

If we use this space variant range function to construct our
steering filter regularization operator we obtain Figure~\ref{fig:fault.interp}.
Instead of creating a smooth change over the fault we see a discontinuity
at the fault boundary at the price of doing a poor estimate of the
velocity to the left of the fault.

\sideplot{fault.interp}{width=3.0in,height=3.0in}{The result of interpolating
the fault model using a space-varying range function.}

Our interpolation fails at the
fault because we are not correctly describing the models covariance  matrix.
Away from the fault a given point as high correlation with the point
surrounding the fault.  At the fault boundary this correlation is shifted.
Steering filters do not the capability of describing this shift as
a result we are left with the choice of either approximating as smooth
change or giving up as we approach the boundary.
