The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 205

Equations ( 9.4 )and( 9.5 ) establish the relationship x a x a . x b ; D y ; R ; B / and

x a in ( 9.17 )intermsof allow the expression of the first order analysis variation ı

x b ;

variations in the DAS input components y ; R ,and B .

9.3.1 Sensitivity to Observations and Background

Baker and Daley ( 2000 ) derived the equations of the forecast sensitivity with respect
to observations and background

@e
@ y D K T @e

(9.19)

@ x a

@e
@

@e
@ x a D @e x a H T @e

I H T K T

x b D Œ

(9.20)

@ @ y

identity matrix. It is noticed that the x b -sensitivity
equation ( 9.20 ) is formally valid only for a linear observation operator, h

n n

where I denotes the

/ D Hx ,
since it neglects the dependence of the linearized observation operator ( 9.3 )on x b .
For the purpose of estimating the forecast sensitivity to background error covariance
parameters we will simply interpret ( 9.20 ) as a vector notation. Second order
derivative information or additional approximations are also necessary to evaluate
the observation sensitivity in a variational DAS with multiple outer loop iterations
( Daescu 2008 ; Tr emolet 2008 ).
The relationship
@e
@

.

x

x a x b

@e
@

x b / H x a x b /

x b ;

R n D

y ;

y h . . (9.21)

R p

may be established from ( 9.19 ), ( 9.20 ), and the analysis equations ( 9.4 ), ( 9.5 ) and its
significance to the parametric error covariance sensitivity is explained in Sect. 9.3.3 .
The evaluation of the observation sensitivity is currently integrated in the routine
activities at NWP centers to monitor the observing system performance using OBSI
measures such as ( 9.9 )and( 9.10 ). As shown below, the vectors ( 9.19 )and( 9.20 )
are also key ingredients to obtain information on the forecast R -and B -sensitivity,
respectively.

9.3.2 Forecast R -and B -Sensitivity and Impact Estimation

The forecast R -and B -sensitivity and impact estimation identifies those error
covariance parameters of potentially high forecast impact and provides guidance
on the forecast benefit that may be achieved from adjusting the error covariance