Geoscience Reference
In-Depth Information
Define a cost function
J
2
.G/
D
1
2
<e.1
W
N/;R
1
.N/e.1
W
N/>
(2.22)
which is an analog of the cost function
J
1
.x.0//
in (
2.5
), where
Nm
Nm
R.N/
D
I
˝
R
2
R
(2.23)
is the ij
th
where
A
˝
B
D
Œa
ij
B
where
a
ij
element of the matrix A, is called
the Kroneker product of
A
and
B
. Clearly,
R.N/
is a block diagonal matrix whose
diagonal blocks are
R
and the off-diagonal blocks are zero matrices. Also, define
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
G
G
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
2
J
p
.G/
D
ˇ
2
(2.24)
F
G
where
is a prior estimate of
G
,
ˇ>0
is a penalty parameter (the larger its value
"
P
i;j
D
1
a
ij
#
2
is called the Frobe-
)and
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
A
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
F
G
the closer the estimate of
G
is to
D
nius matrix norm (which is an extension of the Euclidean norm for the matrix
A
).
Zou et al.
(
1992
)and
Stauffer and Seaman
(
1990
) seek to minimize
Q
1
.G/
D
J
2
.G/
C
J
p
.G/
(2.25)
using the nudged dynamics in (
2.7
) as a strong constraint.
This equality constrained minimization problem can be solved in one of two
ways: using a Lagrangian formulation (
Thacker and Long
(
1988
)) or using the first-
order variational formulation (
Lewis et al.
(
2006
)).
In either approach, the gradient r
G
Q
1
.G/
2
R
n
m
is computed which is then
used in a minimization algorithm to obtain a
.
There are two difficulties associated with the above formulation. First is the
question related to the choice of the prior value
G
that minimizes
Q
1
.G/
G
of the unknown nudging
coefficient. The second and more serious problem is the inherent need to account for
the temporally correlation of the forecast errors
e.1/;e.2/;:::;e.N/
. Exclusion of
G
this correlation introduces a bias in the optimal estimate
of
G
(
Lakshmivarahan
and Lewis
(
2011
)).
In the following we provide a pathway to quantify this inherent temporal
correlation. To this end, first rewrite (
2.7
)as
x.k
C
1/
D
f.x.k/;G/
C
G
z
.k/
(2.26)
or as
x.k
C
1/
D
F.x
k
; x
k
;G/
C
GV.k/
(2.27)
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