Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 241

Fig. 10.8 Ratio of
background ( a )and
observational ( b )cost
functions versus the
theoretical minimum for
overfit ( black )and
constrained ( grey ) methods.
The prior observational error
controls level

a

3 Obs
12 Obs

1

0.8

0.6

0.4

0.2

0

1 2 3 4

Outer Loop

b

3 Obs
12 Obs

1.1

1

0.9

0.8

0.7

0.6

0.5

1 2 3 4

Outer Loop

cost-function minima are functions of the trace of the Kalman gain matrix, and are
given by:

J b

min D 1

2 .N obs

Tr ΠG k 1 K k / (10.16)

J o min D 1

Œ

G k 1 K k

Tr (10.17)

2

Normally, to compute the trace of the matrices involved would be prohibitively
expensive; however, for the Lorenz63 system, the Kalman Gain matrix is computed
explicitly to solve ( 10.16 )and( 10.17 ).
Both methods tend to underestimate the background cost because the actual
background error is less than the prescribed; however, the overfit method increas-
ingly underestimates the background cost with each subsequent outer-loop. This
is well illustrated in Fig. 10.8 (other results are very similar and are not shown).