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0.51
0.44
0.42
60
°
N
0.40
0.38
0.36
0.34
30
°
N
0.32
0.30
0.28
0
°
N
0.27
0.25
0.23
0.21
30
°
S
0.19
0.17
0.15
60
°
S
0.13
0.11
0.09
0.07
150
°
W
120
°
W
°
W
°
W
°
W
°
E
°
E
°
E
°
E
120
°
E
150
°
E
Fig. 4.10
Observation Influence (
OI
) of AMSU-A channel 8 for October 2011. High influential
points are close to 1 and low influential points are close to 0
tropics the maximum
OI
is
0:12
. Since channel 8 observation error variances are
geographically constant the main difference in the observed
OI
pattern is likely due
to the
B
covariance matrix. It looks that the background error correlation are higher
in the tropics than in the extra-tropics.
4.5
Conclusions
The influence matrix is a well-known concept in multi-variate linear regression,
where it is used to identify influential data and to predict the impact on the estimates
of removing individual data from the regression. In this paper the influence matrix
in the context of linear statistical analysis schemes has been derived, as used for
data assimilation of meteorological observations in numerical weather prediction
(
Lorenc 1986
). In particular an approximate method to compute the diagonal
elements of the influence matrix (the self-sensitivities or observation influence)
in ECMWF's operational data assimilation system (4D-Var) has been derived and
implemented. The approach necessarily approximates the solution due to the large
dimension of the estimation problem at hand: the number of estimated parameters
is of the order
25
10
6
.
The self-sensitivity provides a quantitative measure of the observation influence
in the analysis. In robust regression, it is expected that the data have similar self-
sensitivity (sometimes called leverage) - that is, they exert similar influence in
estimating the regression line. Disproportionate data influence on the regression
estimate can have different reasons: First, there is the inevitable occurrence of
incorrect data. Second, influential data points may be legitimately occurring extreme
observations. However, even if such data often contain valuable information, it
is constructive to determine to which extent the estimate depends on these data.
10
9
, and the number of observational data is around
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