Quantifying Observation Impact for a Limited Area Atmospheric Forecast Model - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 124

transfers model space values to observation locations and is the Jacobian matrix
corresponding to the nonlinear forward operator linearized around x b .The

Kalman gain K can be thought of as a weighting matrix and when expanded is
written as,

H. x b /

K D P b H T . HP b H T C R / 1 ;

(6.5)

where P b is the background error covariance matrix and R is the covariance
matrix of the observations. A more detailed derivation of these equations and the
assumptions utilized in the formulation is given in Daley ( 1991 ).
The analysis equation ( 6.4 ) can be rewritten as

x a D x b KHx b C Ky D . I KH / x b C Ky ; (6.6)

with the identity matrix I . Differentiating Eq. 6.6 with respect to y gives an

expression for the sensitivity of the analysis field with respect to the observations,

N N

@ x a

@

y D K T

: (6.7)

Likewise, the sensitivity with respect to the background field is obtained by
differentiating Eq. 6.6 with respect to x b ,

@ x a

T

D I H T K T

x b D .

I KH / : (6.8)

@

The focus of this chapter is gradients with respect to observations, so @ x a
@

will not

x b

be mentioned further.
An expression for the sensitivity of the scalar function in Eq. 6.2 with respect to
the observations is obtained by applying the chain rule for derivatives to Eqs. 6.3
and 6.7 to obtain,

@J
@ y D @J x a @

x a
@ y D K T @J

x a :

(6.9)

@ @

Therefore, the sensitivity of a scalar function of a model's output with respect to
the observations that were used to create the model's analysis field is obtained by
applying the transpose of the Kalman gain to the result of the backward in time
adjoint NWP integration.
Since P b and R are symmetric matrices,

HP b H T C R . / is also symmetric and

Eq. 6.9 can be expanded to give,

@J
@

/ 1 HP b @J
@

HP b H T C R

y D . x a :

(6.10)

An examination of Eq. 6.10 reveals that most of the matrix operations performed
in the original analysis procedure (Eq. 6.4 ) also appear in the gradient calculation,
only in a different order. Therefore, the adjoint of the DA system is obtained by
reordering the routines of the analysis scheme and is much easier to formulate than
the adjoint of the NWP model.