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Kalman Filters Family in Geoscience and Beyond pp. 321-376 $100.00
Authors:  (Olivier Pannekoucke and Christophe Baehr, Meteo-France/CNRS, CNRM/GAME (URA 1357))
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). 

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Kalman Filters Family in Geoscience and Beyond pp. 321-376