The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 208

and ( 9.33 ) may be evaluated prior to the actual implementation of the model R in the
DAS and provide insight on the potential forecast gain at a reduced computational
effort.

9.3.2.2 Forecast B-Sensitivity and Impact Estimation

From ( 9.5 )and( 9.65 ), the first order variation in the K operator associated with a
variation ı B in the background error covariance model B is expressed as

BH T HBH T 1 BH T HBH T 1 H BH T HBH T 1 ı K D ı Œ C R Œ C R ı Œ C R

BH T HBH T 1

D Œ I KH ı Œ C R (9.34)

An explicit relationship between ıe and ı B is obtained by replacing ( 9.34 )in( 9.23 ),

@e
@

BH T HBH T 1 Œ x b / ıe D

x a ;Œ

I KH ı Œ C R y h .

R n

@e
@

T

BH T HBH T 1 Πx b / D ΠI KH

x a ;ı

ΠC R y h . (9.35)

R n

With the aid of ( 9.6 )and( 9.20 ), ( 9.35 ) may be expressed as

@e
@

BH T z ıe D

x b ;ı

(9.36)

R n

From ( 9.36 )and( 9.62 ), the forecast B -sensitivity is the rank-one matrix

H T z T

@e
@

B D @e

2 R n n (9.37)

@

x b

A first order assessment of the forecast performance of a new background error
covariance model

B , as compared with the model B in the DAS, may be obtained

B D B B in ( 9.36 ), by setting ı

/ Œı H T z T

@e
@

x a . B x a . eΠ/ eΠB B (9.38)

x b

Having available the x b -sensitivity vector defined as in ( 9.20 ), the evaluation
of the first order impact estimate ( 9.38 ) may be performed prior to the actual
implementation of the model B in the DAS at a computational cost roughly
equivalent to the cost of a post-multiplication operation ( 9.7 ).