A Question of Adequacy of Observations in Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 115

where the sensitivities and forecast are assumed known and incremental changes to
control are the unknowns. Optimal changes to control are those that minimize the
cost function. In terms of ( 5.6 ), the cost function ( 5.5 ) is rewritten as

X

J.x.0/;;/ D f e. i / (5.7)

i D 1

Π@x.k/

@x.0/ k D i x.0/ Π@x.k/

k D i Π@x.k/
@

k D i g

@

where .

Let us define a sensitivity matrix

e. i / D z . i / x f . i /

S 2 R M 3 :

4 5

Π@x.k/

@x.0/ k D 1 Π@x.k/

k D 1 Π@x.k/
@

k D 1

@

Π@x.k/

@x.0/ k D 2 Π@x.k/

k D 2 Π@x.k/
@

k D 2

S D (5.8)

@

Π@x.k/

@x.0/ k D M Π@x.k/

k D M Π@x.k/
@

k D M

@

E 2 R M 1 :

and an error vector

4 5

z . 1 / x f . 1 /

z . 2 / x f . 2 /

E D (5.9)

z . M / x f . M /

" 2 R 3 1 : and an incremental control vector

4 5

x.0/

" D (5.10)

Then

J D 1

2 .S" E/ T .S" E/

(5.11)

T

where superscript indicates transpose. The necessary condition for minimization

J J "

of is vanishing of the derivative of with respect to the increment of control .

Satisfaction of this condition gives

" D .S T S/ 1 S T E

(5.12)

[See Lewis et al. ( 2006 ) for details].