  Photometric variability of a directly imaged exo-Earth conveys spatial
information on its surface and can be used to retrieve a two-dimensional
geography and axial tilt of the planet (spin-orbit tomography). In this study,
we relax the assumption of the static geography and present a computationally
tractable framework for dynamic spin-orbit tomography applicable to the
time-varying geography. First, a Bayesian framework of static spin-orbit
tomography is revisited using analytic expressions of the Bayesian inverse
problem with a Gaussian prior. We then extend this analytic framework to a
time-varying one through a Gaussian process in time domain, and present
analytic expressions that enable efficient sampling from a full joint posterior
distribution of geography, axial tilt, spin rotation period, and
hyperparameters in the Gaussian-process priors. Consequently, it only takes 0.3
s for a laptop computer to sample one posterior dynamic map conditioned on the
other parameters with 3,072 pixels and 1,024 time grids, for a total of $\sim 3
\times 10^6$ parameters. We applied our dynamic mapping method on a toy model
and found that the time-varying geography was accurately retrieved along with
the axial-tilt and spin rotation period. In addition, we demonstrated the use
of dynamic spin-orbit tomography with a real multi-color light curve of the
Earth as observed by the Deep Space Climate Observatory. We found that the
resultant snapshots from the dominant component of a principle component
analysis roughly captured the large-scale, seasonal variations of the clear-sky
and cloudy areas on the Earth.
