Inverse Solution Methods and Perturbation Estimates for Radiation Leakage
without Scattering

David Ketcheson

The inverse problem of estimating system parameters from measurements of
radiation leakage from a decay gamma source surrounded by a series of
concentric spherical shield layers is considered.  It is assumed that
measured lines are well resolved and that any scattered photons are
removed from the line; therefore, scattering is ignored.  With this
approximation, the transport equation is solved semi-analytically
using ray tracing.  Gradients of the calculated leakage with respect
to the parameters are evaluated either by adjoint differentiation
or by differentiation of the ray trace expression for the leakage.
These gradients are used in gradient-based root-finding and optimization
methods.  The resulting methods are able to estimate unknown source
and shield thicknesses, densities, and compositions using only measured
gamma-ray leakage fluxes.  These methods using semi-analytic gradients are
compared with existing methods using discrete ordinates calculations and
adjoint differentiation.  The ray trace derivatives are more accurate and
computationally efficient, resulting in a faster and more robust inverse
solution.  Perturbation estimates for internal interface perturbations
are considered, and estimates are developed that are highly accurate
over a very large range of perturbations.