Kalman Filters Family in Geoscience and Beyond pp. 321-376
Authors: (Olivier Pannekoucke and Christophe Baehr, M´et´eo-France/CNRS, CNRM/GAME (URA 1357))
Abstract: Being able to predict the weather is one of the greatest challenges of mankind. This success relies on the Kalman filter equations, and its various generalization or approximations. The aims of the chapter is to see why Kalman equations are needed and also to provide various generalization and approximation of information dynamics. At a theoretical level, the atmosphere lies in a particular phase space whose state at time q is denoted by Xq. All the physical process imply a time evolution of this state from q to q + 1 and it is formally written by Xq = ˜ Mq(Xq−1), (1) where ˜Mq corresponds to the propagator underlying to the nature. A numerical weather prediction model is a dynamical system that incorporate all pertinent physical process to provide a worth information toward the forecaster. This corresponds to a set of partial derivative equation that have to be time-integrated from a known state. Of course, this procedure assumes that numerical weather prediction is a deterministic process: one state leads to a one and only one time evolution of the flow (we hope it is). The numerical model can be viewed as a simple non-linear equation that makes evolving the numerical representation of the atmosphere Xq from the time q to the time q + 1 according to Xq =Mq(Xq−1) +Wq, (2) whereMq corresponds to the propagator of the numerical model that differs from the nature propagator ˜ Mq implying to introduce a correction Wq representing a model error due to various approximations of the real physics, for instance the parametrization of the turbulence or of the diphasic process in clouds. Wq is assumed to be a centered Gaussian random variable with covariance matrix Qq. Of course, despite of the complexity of the model and all the very clever things you put inside, to be useful you need to provide the right initial state at time q so to obtain the right time evolution of the real atmosphere (and also before it happens to be expandable, that is the constraint we face).