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PIERRE BRASSEUR
the ocean attractor which cannot be ignored during the assimilation
process. Therefore, it makes sense to renounce making corrections in
the directions where the errors eventually die away because of the attractive nature of the system. Secondly, the state space as defined in
Section 3 is determined by the number of degrees of freedom implied by
the model discretization, which is much larger than the actual number
of dynamical features of interest in the system. Thirdly, less is probably
known about the errors than about the dynamics, and the lack of statistical information will make a full KF superfluous anyway [Cane et al.,
1996], especially in the ocean where the flow of observations accumulates
at a fairly slow rate.
Reducing the dimensionality of the problem can be formulated in
terms of state space or error space with quite dierent implications. A
variety of approaches widely used in oceanography are discussed below.
5.1
Reducing the dimensionality of the state
space
The reduced state can be defined explicitly using a transformation
operator T to convert the full model state x (of dimension n) into a
reduced state w of dimension r < n :
w = Tx
(30)
The statistical properties of x such as error covariance matrices can be
easily transformed in the low-dimension space using (30). Dynamical
equations can be derived for w using a pseudo-inverse of T and the KF
can be entirely reformulated in the low-dimension space, with the condition that the null space associated with (30) be dynamically uncoupled
from the reduced space [Fukumori and Malanotte-Rizzoli, 1995]. By introducing this transformation, the r elements of w become the actual
degrees of freedom of the estimation problem.
Many possibilities exist for defining the transformation, such as a
truncation of the model spectrum or a selection of multivariate modes
of system variability [Cane et al., 1996]. Another approach explored
by Dee [1991] takes advantage of physical relationships between certain
model variables. The reduction of the state space dimension can also
be achieved simply by building the estimation vector from a selection
of model state variables on a coarser grid [Fukumori, 1995] or with dynamical variables closely correlated with the observed signal. It is worth
noting that, for sub-spaces which only preserve the scales larger than
those of the observed signal, an extra term must be added to the observation error covariance matrix R to account for the truncated modes
PIERRE BRASSEUR
the ocean attractor which cannot be ignored during the assimilation
process. Therefore, it makes sense to renounce making corrections in
the directions where the errors eventually die away because of the attractive nature of the system. Secondly, the state space as defined in
Section 3 is determined by the number of degrees of freedom implied by
the model discretization, which is much larger than the actual number
of dynamical features of interest in the system. Thirdly, less is probably
known about the errors than about the dynamics, and the lack of statistical information will make a full KF superfluous anyway [Cane et al.,
1996], especially in the ocean where the flow of observations accumulates
at a fairly slow rate.
Reducing the dimensionality of the problem can be formulated in
terms of state space or error space with quite dierent implications. A
variety of approaches widely used in oceanography are discussed below.
5.1
Reducing the dimensionality of the state
space
The reduced state can be defined explicitly using a transformation
operator T to convert the full model state x (of dimension n) into a
reduced state w of dimension r < n :
w = Tx
(30)
The statistical properties of x such as error covariance matrices can be
easily transformed in the low-dimension space using (30). Dynamical
equations can be derived for w using a pseudo-inverse of T and the KF
can be entirely reformulated in the low-dimension space, with the condition that the null space associated with (30) be dynamically uncoupled
from the reduced space [Fukumori and Malanotte-Rizzoli, 1995]. By introducing this transformation, the r elements of w become the actual
degrees of freedom of the estimation problem.
Many possibilities exist for defining the transformation, such as a
truncation of the model spectrum or a selection of multivariate modes
of system variability [Cane et al., 1996]. Another approach explored
by Dee [1991] takes advantage of physical relationships between certain
model variables. The reduction of the state space dimension can also
be achieved simply by building the estimation vector from a selection
of model state variables on a coarser grid [Fukumori, 1995] or with dynamical variables closely correlated with the observed signal. It is worth
noting that, for sub-spaces which only preserve the scales larger than
those of the observed signal, an extra term must be added to the observation error covariance matrix R to account for the truncated modes