Probabilistic noise reduction

State estimation is an important factor in the production of accurate forecasts. Great effort isexpended in reducing the noise inherent in observations, to produce a “best” estimate of thetrue system state. But noisy observations necessitate a probabilistic, not a deterministic, approach to state estimation. A state’s probabilistic description is rarely Gaussian, and requiresinformation beyond variance magnitude; the correct distribution is restricted by the underlyingstructure of the system attractor. The concepts of finite-time stable and unstable sets are introducedand data assimilation-based methods for their estimation are developed. 4-dimensionalvariational assimilation proves adept at finding the finite-time stable set valid at the beginningof assimilation windows while the ensemble Kalman filter is capable of approximating the finitetimeunstable set at any time that an observation is available. Combining the results of the twoschemes produces a probabilistic estimate of the system state that is superior to either inisolation.