What: Estimate the state of a fluid system - the atmosphere or oceans - from incomplete and inaccurate observations, with the help of dynamical models. When: After the observations have been made and before making a numerical forecast of the system. If the evolution of the system over some finite time is to be evaluated - i.e., if interested in climate rather than prediction - sequential estimation proceeds by scanning through the observations over the interval, forward and back. How: Admit that the dynamical model of the system isn't perfect either. Assign relative weights to the current observations and to the model forecast, based on past observations, that are inversely proportional to their respective error variances. Yes, but: To compute the forecast errors is computationally expensive. So what: Compromise! The thrust of this review is to illustrate some smart ways of (i) near-optimal, but computationally still feasible implementation of the extended Kalman filter (EKF), while using (ii) the EKF for observing system design, as well as for estimating (iii) the state and parameters of (iv) unstable and strongly nonlinear systems, including (v) the coupled ocean-atmosphere system.
