Geoscience Reference
In-Depth Information
Integrating (
2.9
), we get
Z
Z
x.T/
x.0/
D
F.x.t//dt
C
G
f.t/Œx
a
.t/
x.t/dt
(2.12)
0
0
Brill et al.
(
1991
) further postulate that the first integral which is the contribution
of the model accounts for the fraction
.1
˛/Œx.T/
x.0/
of the total change in
x.t/
0
T
the solution
from time
to
asgivenbytheleft-handsideof(
2.12
), where
0<˛<1
. Consequently, the second integral accounts for the remainder of the
change leading to the following relation:
Z
T
aŒx.T/
x.0/
D
G
f.t/Œx
a
.t/
x.t/dt
(2.13)
0
To further simplify the evaluation of the integral on the right-hand side of (
2.13
),
Brill et al.
(
1991
) make one more assumption; namely, the nudged solution
x.t/
varies linearly from
x.0/
D
x
a
.0/
to
x.T/
.Thatis,
x.t/
D
x.0/
C
t
T
Œx.T/
x.0/
(2.14)
Now subtracting (
2.14
) from (
2.11
),
Œx
a
.t/
x.t/
D
t
T
Œx
a
.T/
x
a
.0/
t
T
Œx.T/
x.0/
(2.15)
Substituting (
2.15
)in(
2.13
) and simplifying, we readily obtain
˛ˇ
.1
ˇ/
T
R
T
G
D
(2.16)
0
tf.t/dt
where
ˇ
D
x.T/
x.0/
x
a
.T/
x
a
.0/
D
x.T/
x.0/
x
a
.T/
x.0/
(2.17)
is a fraction of the change in the nudged forecast to that of the analysis. For the case
when
t
T
3
t
T
2
f.t/
D
6:75
C
6:75
(2.18)
4:1
10
4
Brill et al.
(
1991
) in their Appendix provide the values of
G
that range from
2:6
10
3
. They also examine the contour plots of
to
Z
T
G
T
˛ˇ
1
ˇ
tf.t/dt
D
(2.19)
0
in the
˛
ˇ
plane.
Search WWH ::
Custom Search