Geoscience Reference
In-Depth Information
Tabl e 5. 1
Sensitivity functions for the correct control (left-hand columns) where correct control
is given by
x.0/
D
1:0
,
D
11:0
,and
D
2:5
, and for the incorrect control (right-hand columns)
where the incorrect control is given by
x.0/
D
2:0
,
D
10:0
,and
D
3:0
. The bold numbers
indicate errors greater than 40 %
Correct Sensitivity
Incorrect Sensitivity
@x.k/
@x
0
@x.k/
@
@x.k/
@
@x.k/
@x
0
@x.k/
@
@x.k/
@
k
0
1.000
0.000
0.000
1.000
0.000
0.000
1
0.750
0.250
1.000
0.700
0.300
0.800
2
0.562
0.438
1.500
0.490
0.510
1.120
3
0.422
0.578
1.688
0.343
0.657
1.176
4
0.316
0.684
1.688
0.240
0.760
1.098
5
0.237
0.763
1.582
0.168
0.832
0.960
6
0.178
0.822
1.424
0.118
0.882
0.807
7
0.134
0.866
1.246
0.082
0.918
0.659
8
0.100
0.900
1.068
0.058
0.942
0.527
9
0.075
0.925
0.901
0.040
0.960
0.415
10
0.056
0.944
0.751
0.028
0.972
0.323
11
0.042
0.958
0.619
0.020
0.980
0.249
12
0.032
0.968
0.507
0.014
0.986
0.190
13
0.024
0.976
0.412
0.010
0.990
0.144
14
0.018
0.982
0.333
0.007
0.993
0.109
15
0.013
0.987
0.267
0.005
0.995
0.081
16
0.010
0.990
0.214
0.003
0.997
0.061
17
0.008
0.992
0.170
0.002
0.998
0.045
18
0.006
0.994
0.135
0.001
0.998
0.033
X
M
J.x.0/;;/
D
1
2
.
i
/
x.
i
/
2
Œ
z
(5.5)
i
D
1
5.3
Forward Sensitivity Method (FSM) Applied to Air/Sea
Interaction Model
The strong-constraint forecast of air temperature at observation point
i
is given
by
. It is generally different than the observation due to incorrect control
and error in the observation. The basic idea behind FSM is that corrections to
control can annihilate the difference between forecast and observation. To first order
in the Taylor series expansion about the forecasted state, the new estimate of air
temperature at
x
f
.
i
/
i
is given by
x.
i
/
Df
x
f
.
i
/
C
Œ
@x.k/
(5.6)
@x.0/
k
D
i
x.0/
C
Œ
@x.k/
@
k
D
i
C
Œ
@x.k/
@
k
D
i
g
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