  Mountain ranges, volcanoes, trenches, and craters are common on rocky bodies
throughout the Solar System, and we might we expect the same for rocky
exoplanets. With ever larger telescopes under design and a growing need to not
just detect planets but also to characterize them, it is timely to consider
whether there is any prospect of remotely detecting exoplanet topography in the
coming decades. To test this, we devised a novel yet simple approach to detect
and quantify topographical features on the surfaces of exoplanets using transit
light curves. If a planet rotates as it transits its parent star, its changing
silhouette yields a time-varying transit depth, which can be observed as an
apparent and anomalous increase in the photometric scatter. Using elevation
data for several rocky bodies in our solar system, we quantify each world's
surface integrated relief with a "bumpiness" factor, and calculate the
corresponding photometric scatter expected during a transit. Here we describe
the kinds of observations that would be necessary to detect topography in the
ideal case of Mars transiting a nearby white dwarf star. If such systems have a
conservative occurrence rate of 10%, we estimate that the upcoming Colossus or
OWL telescopes would be able to detect topography with <20 hours of observing
time, which corresponds to ~400 transits with a duration of 2 minutes and
orbital period of ~10 hours.
