Recent Applications in Representer-Based Variational Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 291

H T ΠH T C R / 1 D B H T D / 1 ; K . s / D B . s / HB . s / . s . s / . s (12.26)

/ D P B
l D 1

/ D P R
k D 1

T B l … l T R k … k

s l … l s k … k with B . s and R . s . The reduced

J l J k

values for the sub-parts and of the objective function J . s / are

T B

J l . x a . // D d T D 1 H … l / … l H T D 1 d s . s ; (12.27)

with expected value

T

E f J l . x a . // gD s l Œ … l H T D / 1 … l s Tr HB . s / . s ; (12.28)

and

J k . x a . // D Œ … k . y Hx a . // T R .s/ 1 Œ … k . y Hx a . s s s //

(12.29)

T R

D d T D / 1 … k .s/ … k / 1 d . s D . s ;

with expected value

T

E f J k . x a . // gD s k Tr Œ … k R / 1 … k s . s / D . s : (12.30)

The criterion for the tuning parameters is that the relations

J l . x a .

s //

s l D

(12.31)

T

Œ … l / 1 … l

Tr HB . s / H T D . s

and

J k . x a .

s //

s k D (12.32)

T

Œ … k R / 1 … k . / .

Tr s D s

are exactly satisfied. Desroziers and Ivanov ( 2001 ) proposed an iterative approach
( fixed-point algorithm )tosolve( 12.31 )and( 12.32 ), namely,

J l .

x a . s i //

s li C 1 D (12.33)

s i / 1 … l T Œ … l HB Tr . s i / H T D .

J k . x a .

s i //

s ki C 1 D ; (12.34)

T

Œ … k s i / 1 … k Tr R . s i / D .

observing that the first iteration of the fixed-point algorithm gives a good estimate
of the converged results.