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the observation error correlation will slightly increase the observation influence and
for
¢
o
¢
b
the observations will not be more influent in the analysis despite R is
not diagonal.
(i)
R
diagonal and
B
non-diagonal (
˛
¤
0; ˇ
D
0
)
. Equations (
4.24
)and(
4.25
)
reduce respectively to
r
C
1
˛
2
r
2
C
1
˛
2
C
2r
S
11
D
S
22
D
(4.26)
r˛
r
2
C
1
˛
2
C
2r
S
12
D
S
21
D
(4.27)
It can be seen that if the observations are very close compared to the scale-length
of the background error correlation, i.e.
'
1
(data dense area), then
1
r
C
2
S
11
D
S
22
D
S
12
D
S
21
'
(4.28)
Furthermore, if
¢
b
D
¢
o
,thatis
r
D
1
, we have three pieces of information
with equal accuracy and S
11
D
S
22
D
1=3
. The background sensitivity at both
locations is
1
S
11
D
1
S
22
D
2=3
. If the observation is much more accurate
than the background (
¢
b
>> ¢
o
/
,thatis
r
0
, then both observations have
influence S
11
D
S
22
D
1=2
, and the background sensitivities are
1
S
11
D
1
S
22
D
1=2
.
Let's now turn to the dependence on the background-error correlation
'
,forthe
case
¢
b
D
¢
o
.
r D
1/
.Itis
S
11
D
S
22
D
2
˛
2
4
˛
2
(4.29)
˛
4
˛
2
S
12
D
S
21
D
(4.30)
If the locations are far apart, such that
'
0
,thenS
11
D S
22
D
1=2
, the background
sensitivity is also
. It can be concluded that where observa-
tions are sparse, S
ii
and the background-sensitivity are determined by their relative
accuracies (
1=2
and S
12
D S
21
D
0
) and the off-diagonal terms are small (indicating that surrounding
observations have small influence). Conversely, where observations are dense, S
ii
tends to be small, the background-sensitivities tend to be large and the off-diagonal
terms are also large.
It is also convenient to summarize the case
r
¢
b
D
¢
o
.r
D
1/
by showing the
projected analysis at location
1
.2
˛
2
/y
1
C
2x
1
˛.x
2
y
2
/
1
4
˛
2
y
1
D
(4.31)
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