  Data assimilation algorithms are used to estimate the states of a dynamical
system using partial and noisy observations. The ensemble Kalman filter has
become a popular data assimilation scheme due to its simplicity and robustness
for a wide range of application areas. Nevertheless, the ensemble Kalman filter
also has limitations due to its inherent Gaussian and linearity assumptions.
These limitations can manifest themselves in dynamically inconsistent state
estimates. We investigate this issue in this paper for highly oscillatory
Hamiltonian systems with a dynamical behavior which satisfies certain balance
relations. We first demonstrate that the standard ensemble Kalman filter can
lead to estimates which do not satisfy those balance relations, ultimately
leading to filter divergence. We also propose two remedies for this phenomenon
in terms of blended time-stepping schemes and ensemble-based penalty methods.
The effect of these modifications to the standard ensemble Kalman filter are
discussed and demonstrated numerically for two model scenarios. First, we
consider balanced motion for highly oscillatory Hamiltonian systems and,
second, we investigate thermally embedded highly oscillatory Hamiltonian
systems. The first scenario is relevant for applications from meteorology while
the second scenario is relevant for applications of data assimilation to
molecular dynamics.
