  In this paper we consider sweeping preconditioners for time harmonic wave
propagation in stratified media, especially in the presence of reflections. In
the most famous class of sweeping preconditioners Dirichlet-to-Neumann
operators for half-space problems are approximated through absorbing boundary
conditions. In the presence of reflections absorbing boundary conditions are
not accurate resulting in an unsatisfactory performance of these sweeping
preconditioners. We explore the potential of using more accurate
Dirichlet-to-Neumann operators within the sweep. To this end, we make use of
the separability of the equation for the background model. While this improves
the accuracy of the Dirichlet-to-Neumann operator, we find both from numerical
tests and analytical arguments that it is very sensitive to perturbations in
the presence of reflections. This implies that even if accurate approximations
to Dirichlet-to-Neumann operators can be devised for a stratified medium,
sweeping preconditioners are limited to very small perturbations.
