Geoscience Reference
In-Depth Information
19.3.2
Correlated Observation Errors
The number of satellite radiance observations can dramatically increase when
cloudy radiances are included. This immediately opens several new data assim-
ilation issues such as observation error correlations and computational overhead.
Consider a commonly used observation equation
y
D
h.x/
C
"
(19.14)
Where
"
is a Gaussian random variable
N.0;R/
and
R
D
˝
""
T
˛
(19.15)
The observation error covariance matrix
is typically defined as diagonal, implying
uncorrelated observation increments. This assumption greatly simplifies data assim-
ilation that requires the inverse of
R
, and is also relatively accurate if observations
are not very close to each other. However, when observations are densely distributed
the uncorrelated observation error assumption may not be justified. The ultimate
consequence of correlated observation errors is that the information content of
near-by observations is reduced compared to their independent information. This
intuitive conclusion can be formalized using mathematical framework of Shannon
information theory (
Shannon and Weaver 1949
; also Appendix 2). Let
R
Y
1
and
Y
2
represent random observation errors for two near-by observations. Using a
general relationship between entropy
H
and mutual information
I
(e.g.,
Cover and
Thomas 2006
)
I.Y
1
I
Y
2
/
D
H.Y
1
/
C
H.Y
2
/
H.Y
1
;Y
2
/
(19.16)
as well as (
19.34
) from Appendix 2
I.Y
1
I
Y
2
/
D
I.Y
1
I
Y
1
/
C
I.Y
2
I
Y
2
/
H.Y
2
;Y
2
/:
(19.17)
Since by definition
H.Y
1
;Y
2
/
0
for arbitrary random variables
Y
1
and
Y
2
,we
have
I.Y
1
I
Y
2
/
I.Y
1
I
Y
2
/
C
I.Y
2
I
Y
2
/:
(19.18)
The relation (
19.18
) states that mutual information of dependent variables is smaller
than mutual information of independent variables. Since in Gaussian framework the
notion of dependence is directly related to correlations, one can say that correlated
observations bring less information than uncorrelated observations. This conclusion
implies a need to account for correlated observation in all-sky satellite radiance
assimilation.
There are several possible ways one could address observation error correlations
in data assimilation:
1.
Increase observation errors
: If observation density is high, reduce the impact of
dense observations by increasing the observation error.
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