  Filtering is concerned with the sequential estimation of the state, and
uncertainties, of a Markovian system, given noisy observations. It is
particularly difficult to achieve accurate filtering in complex dynamical
systems, such as those arising in turbulence, in which effective
low-dimensional representation of the desired probability distribution is
challenging. Nonetheless recent advances have shown considerable success in
filtering based on certain carefully chosen simplifications of the underlying
system, which allow closed form filters. This leads to filtering algorithms
with significant, but judiciously chosen, model error. The purpose of this
article is to analyze the effectiveness of these simplified filters, and to
suggest modifications of them which lead to improved filtering in certain
time-scale regimes. We employ a Markov switching process for the true signal
underlying the data, rather than working with a fully resolved DNS PDE model.
Such Markov switching models haven been demonstrated to provide an excellent
surrogate test-bed for the turbulent bursting phenomena which make filtering of
complex physical models, such as those arising in atmospheric sciences, so
challenging.
