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a=0.9, b=0
a=0.9, b=0.9
a=0, b=0
a=0, b=0.9
1
0.8
0.6
0.4
0.2
0
r
Fig. 4.1
Self-Sensitivities or Observation Influence (
OI
) as a function of the ratio between the
observation error variance and the background error variance. Four different cases are shown:
highly correlated
R
and uncorrelated
B
(
thick black line
). Highly correlated
R
and highly correlated
B
(
thick grey line
). Uncorrelated
R
and highly correlated
B
(
thin grey line
). Uncorrelated
R
and
uncorrelated
B
(
dashed black line
)
b
o
.1
˛ˇ/
C
b
.1
˛
2
/
b
.1
˛
2
/
C
o
.1
ˇ
2
/
C
2
b
o
.1
˛ˇ/
S
11
D
S
22
D
(4.22)
b
o
.˛
ˇ/
b
.1
˛
2
/
C
o
.1
ˇ
2
/
C
2
b
o
.1
˛ˇ/
S
12
D
S
21
D
(4.23)
r
D
o
=
b
For
'
¤˙
1
and
“
¤˙
1
(
R
and
B
are full rank matrices). Let's define
,
(
4.22
)and(
4.23
) reduce to
r.1
˛ˇ/
C
1
˛
2
r
2
.1
ˇ
2
/
C
1
˛
2
C
2r.1
˛ˇ/
S
11
D
S
22
D
(4.24)
r.˛
ˇ/
r
2
.1
ˇ
2
/
C
1
˛
2
C
2r.1
˛ˇ/
S
12
D
S
21
D
(4.25)
Figure
4.1
shows the diagonal elements of the influence matrix as a function
of
(
4.24
). From now on, S
ii
is also indicated as Observation Influence
(
OI
). In general, the observation influence decreases with the increase of
r;
S
ii
D S
ii
.r/
r
.For
highly correlated (
'
D
0:9; “
D
0:9
)
R
and
B
and diagonal (
'
D
0; “
D
0
)
R
and
B
, the observation influence as a function of
is the same (solid grey line and
dash thick line, respectively). Maximum observation influence is achieved when
B
is diagonal (
r
) (thin black line).
The observation influence will constantly decrease from the 'maximum curve' with
the decrease of the correlation degree in
R
(
B
still diagonal). And the minimum
observation influence curve is achieved when
R
is diagonal (
'
D
0
)and
R
is highly correlated (
“
D
0:9
“
D
0
)and
B
is highly
correlated (
) (thick solid line). It is worth to notice that if the observation
error variance is larger than the background error variance (
'
D
0:9
¢
o
>¢
b
/
introducing
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