Many contemporary problems in science involve making predictions based on partial
observation of extremely complicated spatially extended systems with many degrees
of freedom and with physical instabilities on both large and small scale. Various
new ensemble Kalman filtering strategies have been developed for these applications
and new mathematical issues arise. Recently, Majda et al. developed explicit off-line
test criteria for stable accurate discrete filtering (PNAS 2006, PNAS 2007) and
various reduced Fourier domain Kalman filters for spatially extended turbulent
systems (JCP 2008). Here we extend the analysis and explore the filter skill to the
situation of sparse irregular (non-equispaced) observations, in particular for the prediction
of large-scale singular events. By using the relative entropy and the Shannon entropy
difference to measure the information content we develop new strategies for removing
redundant observations and positioning optimal targeted observations.
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