Chaos, observability and symplectic structure in optimal estimation

Observation, estimation and prediction are universal challenges that become especially difficult when the system under consideration is dynamical and chaotic. Chaos injects dynamical noise into the estimation process that must be suppressed to satisfy the necessary conditions for success: namely, synchronization of the estimate and the observed data. The ability to control the growth of errors is constrained by the spatiotemporal resolution of the observations, and often exhibits critical thresholds below which the probability of success becomes effectively zero. This thesis examines the connections between these limits and basic issues of complexity, conditioning, and instability in the observation and forecast models. The results suggest several new ideas to improve the collaborative design of combined observation, analysis, and forecast systems. Among these, the most notable is perhaps the fundamental role that symplectic structure plays in the remarkable observational efficiency of Kalman-based estimation methods.