  We extend observability metrics based on the empirical observability Gramian
from deterministic nonlinear systems to nonlinear stochastic systems in order
to capture the impact of process noise on observability. We demonstrate that
the empirical observability Gramian can be used to provide an equivalent
condition for a definition of stochastic observability on linear systems, and
that the Gramian can be used to extend stochastic observability to nonlinear
stochastic systems. We further demonstrate through simulation that
consideration of process noise can reveal observability in systems that would
be considered unobservable using traditional deterministic tools.
