  Planned missions will spatially resolve temperate terrestrial planets from
their host star. Although reflected light from such a planet encodes
information about its surface, it has not been shown how to establish surface
characteristics of a planet without assuming known surfaces to begin with. We
present a re-analysis of disk-integrated, time-resolved, multiband photometry
of Earth obtained by the Deep Impact spacecraft as part of the EPOXI Mission of
Opportunity. We extract reflectance spectra of clouds, ocean and land without a
priori knowledge of the numbers or colors of these surfaces. We show that the
inverse problem of extracting surface spectra from such data is a novel and
extreme instance of spectral unmixing, a well-studied problem in remote
sensing. Principal component analysis is used to determine an appropriate
number of model surfaces with which to interpret the data. Shrink-wrapping a
simplex to the color excursions of the planet yields a conservative estimate of
the planet's endmember spectra. The resulting surface maps are unphysical,
however, requiring negative or larger-than-unity surface coverage at certain
locations. Our "rotational unmixing" supersedes the endmember analysis by
simultaneously solving for the surface spectra and their geographical
distributions on the planet, under the assumption of diffuse reflection and
known viewing geometry. We use a Markov Chain Monte Carlo to determine best-fit
parameters and their uncertainties. The resulting albedo spectra are similar to
clouds, ocean and land seen through a Rayleigh-scattering atmosphere. This
study suggests that future direct-imaging efforts could identify and map
unknown surfaces and clouds on exoplanets.
