  Hybrid Monte-Carlo (HMC) sampling smoother is a fully non-Gaussian
four-dimensional data assimilation algorithm that works by directly sampling
the posterior distribution formulated in the Bayesian framework. The smoother
in its original formulation is computationally expensive due to the intrinsic
requirement of running the forward and adjoint models repeatedly. Here we
present computationally efficient versions of the HMC sampling smoother based
on reduced-order approximations of the underlying model dynamics. The schemes
developed herein are tested numerically using the shallow-water equations model
on Cartesian coordinates. The results reveal that the reduced-order versions of
the smoother are capable of accurately capturing the posterior probability
density, while being significantly faster than the original full order
formulation.
