The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 209

9.3.3 Forecast Sensitivity to Error Covariance Parameters

While the explicit evaluation and storage of the R -and B -sensitivity matrices is not
feasible in an operational system, from ( 9.29 )and( 9.37 ) it is noticed that evaluation
and storage of only a few vectors are necessary to capture the information content of
the sensitivity matrices. Of practical significance is the ability to evaluate directional
derivatives associated with perturbations

and to obtain sensitivities to key
parameters used to model the error covariances. The observation sensitivity vector
( 9.19 ) is a key ingredient to both R -and B -sensitivity estimation and techniques
to observation sensitivity and impact estimation in an ensemble-based DAS have
been also formulated ( Liu and Kalnay 2008 ; Liu et al. 2009 ). The R -and B -
sensitivity equations provided in this work are thus of relevance to both variational
and ensemble-based data assimilation systems and their use to perform parameter
sensitivity analysis is presented below.

.ı R ;ı B /

9.3.3.1 Sensitivity to Multiplicative Error Covariance Parameters

A practical approach to perform error covariance tuning relies on the parametric
representation

.s b / D s b B R i .s i / D s i

B ; R i ;i 2 I (9.39)

s b >0 s i >0

where are scalar coefficients used to adjust the weight given

in the DAS to the background information and to the information provided by
the observing system component y i ;i 2 I

and

, respectively ( Chapnik et al. 2006 ;
Desroziers et al. 2009 ). In the formulation ( 9.39 ) it is assumed that f y i ;i 2 I g
is a partition of the observations consisting of data subsets y i 2 R p i ;i 2 I

,
with uncorrelated observation errors such that the model R is structured as a block
diagonal matrix

R i 2 R p i p i R D diag. R i /; (9.40)

The covariance specification . R ; B / in the reference DAS corresponds to all weight

s i D 1; i 2 I s b D 1 parameters in set to 1 i.e., and .From( 9.39 ), the covariance

ıs i ıs b in the weight coefficients are variations induced by perturbations and

expressed respectively, as

R i D ıs i

ı R i ;i 2 I (9.41)

B D ıs b B ı (9.42)

By replacing ( 9.41 )in( 9.28 ) and with the aid of ( 9.30 ), the first order variation in
the forecast aspect x a / ıs i e. is expressed in terms of as

@e
@

ıe D X
i 2 I

ıs i x b / H x a x b / i

y i ;Œ

y h . . (9.43)

R p i