Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 240

Fig. 10.6 P p )of

the posterior error to the
prior projected in data-space.
The overfit ( black ) method
generates a higher error
estimate compared to the
constrained ( grey ) method
across all choices of
background and observational
errors as well as over varying
numbers of observations

Ratio ( P a =

3 Obs
12 Obs

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

Fig. 10.7 Comparison of
prior R and posterior R a
observational error for overfit
( black ) and constrained ( grey )
methods

3

2.5

2

1.5

1

0.5

0

0 0.5 1 1.5 2 2.5 3

R

the observations. The constrained method slightly underestimates the observational
error, but is consistent with the prior . This exposes the overfit method for placing
too much emphasis on the observations.
As a final comparison, the theoretical cost-function minimum is compared to the
minimized cost-function at the end of each outer-loop. Because the overfit method
is unconstrained by P after the first outer-loop, it was shown that it will minimize
J o at the expense of J b .

Bennett ( 2002 ) showed that for correctly specified P and R , the minimum value
of the cost-function is . This measure has been used as a useful

diagnostic by Weaver et al. ( 2003 )and Powell et al. ( 2008 ). Unfortunately, this
measure does not quantify the contribution of each component of the cost function.
Moore et al. ( 2011a ) provides a concise review of the work of Talag r an d ( 1999 ),
Chapnik et al. ( 2006 ), and Desroziers et al. ( 2009 ) along with the derivation for
determining the minimum theoretical values of each cost-function component. The

J min D N obs =2