Geoscience Reference
In-Depth Information
where
f.x;G/
D
M.x/
Gh.x/
(2.28)
and
F.x; x;G/
D
f.x;G/
C
Gh.x/
(2.29)
which is separable in
x
and
x
.
is a vector white noise Gaussian process, it readily follows from
(
2.27
)thatf
x.k/
g
k
0
Since
V.k/
is a first-order nonlinear auto-regressive process of order
1(
Hamilton
(
1994
)). Thus, given
M.x/;M.x/;h.x/;x.0/
and
x.0/
,
x.k/
is a
function of
G
and the complete history noise sequence
V.1/;V.2/;:::;V.k/
for
k
1
Assuming
h.x/
D
x
, the error in (
2.6
) namely
e.k/
D
z
.k/
x.k/
(2.30)
depends on
G
and the noise vector
V.1
W
k/
D
V
T
.1/;V
T
.2/;:::;V
T
.k/
T
2
R
km
(2.31)
V
2
R
Nm
Nm
such that
Consequently, there exists a covariance matrix
.e.j/;e.j//
2
R
m
m
V
ij
D cov
(2.32)
for all
1
i;j
N:
Now define
J
3
.G/
D
1
2
<e.1
W
N/;V
1
e.1
W
N/>
(2.33)
which is a modified version of
J
2
.G/
in (
2.22
). Accordingly, the correct formulation
is as follows: Find
G
that minimizes
Q
2
.J/
D
J
3
.G/
C
J
p
.G/
(2.34)
instead of
in (
2.25
).
We hasten to add that while (
2.34
) is the correct formulation of the optimal nudg-
ing problem, it is very difficult to compute the elements of the covariance matrix
Q
1
.G/
V
for the case when the state transition map
in (
2.7
) is nonlinear. However, when
the dynamics is linear, we can give an explicit expression for the elements of
M
V
that
captures the underlying correlation structure of the forecast errors.
Example 2.1.
Linear Dynamics and Observati
ons
Consider t
he
special case when
M.x/
D
Mx
M.x/
D
Mx
h.x/
D
Hx
,
,
where
M
2
R
n
n
,
M
2
R
n
n
,and
H
2
R
m
n
. Then the observation equation (
2.6
)
becomes
.k/
D
H x.k/
C
V.k/
z
(2.35)
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