We address the recovery of time dependent absorption and scattering coefficients in
the Riemannian transport equation from the albedo operator. Given
,
, a
compact domain with smooth boundary, equipped with a Riemannian metric
, we
first prove the unique determination of a time dependent absorption coefficient in a
subset of the domain of interest, provided that it is known outside this subset. We
then show that we can recover the coefficient in a larger region (and eventually in the
entire domain) by enlarging the data set. Next, we present a uniqueness result for the
reconstruction of the scattering parameter based on the knowledge of the albedo
operator. The proof is based on geometric optics solutions and inversion of
the light ray transform on static Lorentzian manifolds, assuming that the
Lorentzian manifold is a product of a time interval with a simple Riemannian
manifold.
Keywords
inverse problems, transport equation, Riemannian manifold,
Lorentzian manifold, albedo operator, light ray transform
