Geoscience Reference
In-Depth Information
•
Propagate all the sigma points through the nonlinear dynamic and the output
equations,
z
i
D
f.
i
;0/
g
i
D
h.
i
;0/; i
D
1;2;:::;n
I
•
Calculate the mean (prediction) of the state and output,
P
2n
i
D
1
1
2n
x
k
D
z
i
P
2n
1
2n
i
D
1
g
i
I
y
k
D
•
The prediction of the covariance matrices are given by,
P
2n
i
D
1
.
P
x
k
D
1
2n
z
i
x
k
/.
z
i
x
k
/
T
P
2n
P
y
k
D
1
2n
i
D
1
.g
i
y
k
/.g
i
y
k
/
T
P
2n
i
D
1
.
P
x
k
D
1
2n
z
i
x
k
/.g
i
y
k
/
T
P
xx
k
P
yy
k
P
xy
k
x
k
,
Once the prediction of
,
and
are available, the update is given by
x
k
D
x
k
C
K.y
k
y
k
/
where
K
D
P
x
k
ΠP
y
k
1
P
x
k
D
P
x
k
K P
x
k
K
T
:
While UKF avoids the computation of linearization, it requires the integration of
2n
trajectories. For nonlinear systems with a moderate dimension, the UKF is an
reliable and efficient filter for real-time estimation. However, it is not clear if the
idea is applicable to large scale systems with tens of thousands or even millions
of dimensions. For systems with very high dimensions, such as the models for
numerical weather forecast, currently popular approaches include the ensemble
Kalman filter (EnKF) and 4D-Var estimation and prediction (
Anderson 2003
;
Evensen 2007
,
1994
;
Houtekamer and Mitchell 1998
;
Chua and Bennett 2001
;
Courtier et al. 1994
;
Rabier et al. 2000
;
Xu et al. 2005
). These methods are primarily
developed and widely used in the data assimilation community. They are extensively
addressed in the other chapters of this topic. Therefore, we skip the details on EnKF
and 4D-Var methods.
1.4.2
H
1
Filter
The Kalman filter is optimal in a stochastic sense. However, the probability model of
disturbances may not be available for a given system. In this case, one may assume
that the noises are not stochastic but unknown
L
2
functions. The goal of
H
1
filters
is to estimate the state variables in such a way that the gain from noise to estimation
error is as small as possible. Following
Krener
(
2004
), consider a system
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