  The inverse radiative transfer problem finds broad applications in medical
imaging, atmospheric science, astronomy, and many other areas. This problem
intends to recover the optical properties, denoted as absorption and scattering
coefficient of the media, through the source-measurement pairs. A typical
computational approach is to form the inverse problem as a PDE-constraint
optimization, with the minimizer being the to-be-recovered coefficients. The
method is tested to be efficient in practice, but lacks analytical
justification: there is no guarantee of the existence or uniqueness of the
minimizer, and the error is hard to quantify. In this paper, we provide a
different algorithm by levering the ideas from singular decomposition analysis.
Our approach is to decompose the measurements into three components, two out of
which encode the information of the two coefficients respectively. We then
split the optimization problem into two subproblems and use those two
components to recover the absorption and scattering coefficients separately. In
this regard, we prove the well-posedness of the new optimization, and the error
could be quantified with better precision. In the end, we incorporate the
diffusive scaling and show that the error is harder to control in the diffusive
limit.
