We present the first results in a new program intended to make the best use of
all available technologies to provide an effective understanding of waves from
inspiralling black hole binaries in time for imminent observations. In
particular, we address the problem of combining the close-limit approximation
describing ringing black holes and full numerical relativity, required for
essentially nonlinear interactions.  We demonstrate the effectiveness of our
approach using general methods for a model problem, the head-on collision of
black holes. Our method allows a more direct physical understanding of these
collisions indicating clearly when non-linear methods are important.  The
success of this method supports our expectation that this unified approach will
be able to provide astrophysically relevant results for black hole binaries in
time to assist gravitational wave observations.