Observation Influence Diagnostic of a Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 91

frequentist approach, based on a statistical linear analysis scheme providing the Best
Linear Unbiased Estimate ( Talagrand 1997 )of x ,given y and x b . The optimal GLS
solution to the analysis problem (see Lorenc 1986 ) can be written

x a D Ky C . I n KH / x b (4.11)

The vector x a is the 'analysis'. The gain matrix K ( ) takes into account the

respective accuracies of the background vector x b and the observation vector y as
defined by the

n m

n n covariance matrix B and the m m covariance matrix R , with

B 1 C H T R 1 H / 1 H T R 1 K D . (4.12)

Here, H is a matrix interpolating the background fields to the observation

locations, and transforming the model variables to observed quantities (e.g. radiative
transfer calculations transforming the models temperature, humidity and ozone into
brightness temperatures as observed by several satellite instruments). In the 4D-Var
context introduced below, H is defined to include also the propagation in time of the
atmospheric state vector to the observation times using a forecast model.
Substituting ( 4.12 )into( 4.11 ) and projecting the analysis estimate onto the
observation space, the estimate becomes

m n

y D Hx a D HKy C . I m HK / Hx b (4.13)

It can be seen that the analysis state in observation space ( Hx a )isdefinedasasum
of the background (in observation space, Hx b ) and the observations y , weighted by
the square matrices I HK and HK ; respectively.

Equation ( 4.13 ) is the analogue of ( 4.1 ), except for the last term on the right hand
side. In this case, for each unknown component of H x , there are two data values:
a real and a 'pseudo' observation. The additional term in ( 4.13 ) includes these
pseudo-observations, representing prior knowledge provided by the observation-
space background Hx b .From( 4.13 )and( 4.4 ), the analysis sensitivity with respect
to the observations is obtained

m m

S D @ y
@ y D K T H T

(4.14)

Similarly, the analysis sensitivity with respect to the background (in observation
space) is given by

@ y

D I K T H T D I m S (4.15)

@. Hx b /

Let's focus here on the expressions ( 4.14 )and( 4.15 ). The influence matrix for the
weighted regression DA scheme is actually more complex (see Appendix), but it
obscures the dichotomy of the sensitivities between data and model in observation
space.