Observation Influence Diagnostic of a Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 92

The (projected) background influence is complementary to the observation
influence. For example, if the self-sensitivity with respect to the th observation

is S ii , the sensitivity with respect the background projected at the same variable,
location and time will be simply

i

1 S ii . It also follows that the complementary

trace, tr , is not the df for noise but for background, instead.

That is the weight given to prior information, to be compared to the observational
weight tr( S ). These are the main differences with respect to standard LS regression.
Note that the different observations can have different units, so that the units of the
cross-sensitivities are the corresponding unit ratios. Self-sensitivities, however, are
pure numbers (no units) as in standard regression. Finally, as long as R is diagonal,
( 4.6 ) is assured (see Sect. 4.3.2 ), but for more general non-diagonal R -matrices it is
easy to find counter-examples to that property.
Inserting ( 4.12 )into( 4.14 ), we obtain

. I S / D m tr . S /

S D R 1 H B 1 C H T R 1 H / 1 H T . (4.16)

/ 1 is equal to the analysis error covariance matrix A , we can
also write S D R 1 HAH T .

B 1 C H T R 1 H As .

4.3.2 R Diagonal

In this section it is shown that as long as R is diagonal ( 4.6 ) is satisfied. Equation
( 4.16 ) can be written as

S D R 1 H B BH T . HBH T C R / 1 HB H T Π(4.17)

D R 1 HBH T R 1 HBH T . HBH T C R / 1 HBH T

HBH T ,( 4.17 ) becomes Let's introduce the matrix V D

S D R 1 V R 1 V / 1 V . V C R

D R 1 V

/ 1 V

Œ

I .

V C R

D R 1 V

/ 1 . / 1 V

Œ. / .

V C R V C R V C R

D R 1 V

/ 1 R

.

V C R

D R 1 Œ.

/ 1 R / 1 /. .

V C R V C R V C R R (4.18)

D R 1 Œ

/ 1 .

I R V C R R

/ 1 R

D I . V C R

/ 1 V

D . V C R