Geoscience Reference
In-Depth Information
Acknowledgements
We gratefully acknowledge support from the Office of Naval Research
through PE-0601153N.
Appendix 1: Matrix and Notation Definitions
The prior error covariance matrix is extended to include the higher moments:
"
P
f
#
Z Z
T
K
1
T
f
P
f
D
D
;
(7.44)
T
f
F
f
p
f
p
f
whose square root form may be approximated with an ensemble as
©
1
©
2
©
K
Z
D
(7.45)
©
1
˝ ©
1
p
f
©
2
˝ ©
2
p
f
©
K
˝ ©
K
p
f
N
2
-vector constructed
and the vectorized covariance matrix,
p
f
D
v
ec.
P
f
/
,isan
from the concatenation of the
columns of
P
f
and whose organization follows
that of the Kronecker product “˝”, and
N
is the ensemble size.
The extended observation operator takes the following form:
K
H
H
2
H
H
˝
H
H
D
D
:
(7.46)
Note that for nonlinear observation operators one would not use a linearized form of
the operator. Instead, the correct procedure is to operate the nonlinear observation
operator on each member of the ensemble and then perform linear or nonlinear
regression on this new distribution of predicted prior observations against the state
variables needing update (
Houtekamer and Mitchell 2001
).
The covariance matrices in the extended state-space takes the form:
˝
©
f
v
0
T
˛
D
P
f
H
T
T
f
H
2
;
(7.47)
D
vv
T
E
˝
v
0
v
0
T
˛
D
h
v
i
˝
v
T
˛
;
(7.48)
2
4
3
5
HP
f
H
T
C
R
HT
f
H
2
D
vv
T
E
H
2
T
f
H
T
H
2
F
f
H
2
C
A
C
B
C
C
C
R
4
D
(7.49)
:
:
:
:
:
:
:
:
:
2
4
3
5
;
00
0
˝
v
2
˛˝
v
2T
˛
:
:
:
h
v
i
˝
v
T
˛
D
(7.50)
:
:
:
:
:
:
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