METEOROLOGICAL DATA PERSPECTIVE
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where B is the covariance matrix of the background error H b and R the
covariance matrix of the observation error H r which includes the instrument
error and the representativeness error.
The solution in this linear case is given by
x a = x b +K(y-Hx b )
(4)
with the gain matrix K
K=BH
T
(HBH
T + R)
-1
.
(5)
The corresponding analysis error is given by
H a = (I-KH) H b + KH r .
(6)
The analysis error covariance is
A=(I-KH)B=(B
-1 +H
T
R
-1 H)
-1
.
(7)
This is the Optimal least-square estimator, which leads to the minimum
variance for the analysis error, or BLUE = Best Linear Unbiased Estimator.
If all errors are Gaussian, then it is also the maximum likelihood estimate.
When one wants to evaluate the gain brought by the observations, a pure
data count can be misleading. If practically feasible, the computation of the
analysis error covariance A and its comparison with the background error
covariance matrix B will indicate how much benefit was brought by the
observations, in terms of decreasing the error covariance of the estimation of
the atmospheric state. Another approach is to compute the sensitivity of the
analysis with respect to the observations. This leads to estimating the
information content of data types. For example, one can compute the DFS =
Degrees of Freedom for Signal (Rodgers, 2000)
DFS= Tr(KH)
(8)
where the trace of the matrix KH quantifies the gain in information brought
by the observations. As shown by
Hx a = (I-HK) Hx b + HKy
(9)
The HK matrix quantifies the sensitivity of the analysis to the
observations