  In this paper, we investigate the recovery of the absorption coefficient from
boundary data assuming that the region of interest is illuminated at an initial
time. We consider a sufficiently strong and isotropic, but otherwise unknown
initial state of radiation. This work is part of an effort to reconstruct
optical properties using unknown illumination embedded in the unknown medium.
  We break the problem into two steps. First, in a linear framework, we seek
the simultaneous recovery of a forcing term of the form $\sigma(t,x,\theta)
f(x)$ (with $\sigma$ known) and an isotropic initial condition $u_{0}(x)$ using
the single measurement induced by these data. Based on exact boundary
controllability, we derive a system of equations for the unknown terms $f$ and
$u_{0}$. The system is shown to be Fredholm if $\sigma$ satisfies a certain
positivity condition. We show that for generic term $\sigma$ and weakly
absorbing media, this linear inverse problem is uniquely solvable with a
stability estimate. In the second step, we use the stability results from the
linear problem to address the nonlinearity in the recovery of a weak absorbing
coefficient. We obtain a locally Lipschitz stability estimate.
