The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 206

Fig. 9.2 Flow chart of the
relationship between the
forecast aspect and various
input components of the data
assimilation system

o

Σ

y

R

o

a

C

a )

K x e ( x

b

Σ

b

B x

b

C

parameters. The DAS operator K incorporates both R and B models and from ( 9.4 )
the analysis variation induced by a perturbation in the K operator is expressed as

x a D ı x b / ı K Œ y h . (9.22)

By replacing ( 9.22 )in( 9.17 ), the first order forecast variation ıe is expressed in

terms of ı K as

@e
@

x b /

ıe D

x a ;ı

K Πy h . (9.23)

R n

From ( 9.23 )and( 9.62 ), the forecast sensitivity to the K operator is the rank-one
matrix

@e
@ K D @e

x b /

T

2 R n p

x a Œ

y h . (9.24)

@

To obtain the forecast sensitivity to the error covariance specification it is necessary
to analyze how the K operator responds to variations in R and B . Additionally, each
covariance model incorporates a diagonal matrix whose entries are the values

assigned to the error standard deviation and an error correlation model C ,

˙

R D ˙ o C o ˙ o ; B D ˙ b C b ˙ b (9.25)

A flow chart of the functional dependence of the forecast aspect on various DAS
input components is illustrated in Fig. 9.2 and the extension of the adjoint-DAS
applications to parameters in the error covariance specification is discussed next.

9.3.2.1 Forecast R-Sensitivity and Impact Estimation

From ( 9.5 )and( 9.65 ), the first order variation in the K operator induced by a
variation ı R in the observation error covariance model R is expressed as

K D BH T HBH T 1 ı HBH T 1 D K HBH T 1 (9.26) ı Œ C R R Œ C R ı R Œ C R