Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 239

from the true initial conditions in order to better represent the noisy observations.
In fact, comparing the true observations without random error perturbations to the
final analysis trajectories, the overfit method only reduces the innovation residuals
(Eq. 10.3 ) between the model and true observations by 37 %, while the constrained
method reduces it by 88 %. Without a proper background constraint, the overfit
method attempts to best fit the noisy observations. The constrained method attempts
to best represent the noisy observations while decreasing the initial error. This
initial constraint actually provides for a better observational fit against the true
observations.
Another important measure is the consistency between the prescribed prior
background ( P ) and observational ( R ) errors with the posterior estimates from the
analysis. As shown in Desroziers et al. ( 2005 ) and used in Moore et al. ( 2011a ),
two relationships are found. First, two vectors are defined: d b / i D H . . x k .t i //

H . th outer-loop and provides

the difference between the analysis and background at the observation locations;
and,

x b .t i // ,where x k is the integrated solution of the k

d o / i D y i H .

provides the difference between the analysis and the
observations. Desroziers et al. ( 2005 ) shows that the posterior error estimates are
given by,

. x k .t i //

E d b d T D

G k PG k (10.14)

E d o d T D

R : (10.15)

Because each term is computed directly in the Lorenz63 problem, the specified
prior error values are compared with ( 10.14 )and( 10.15 ). First, the posterior error
estimate from the analysis, P a D d b d T , is compared with the diagonal of the prior

error, P p D G k PG k . Because the diagonal values of P are prescribed, one would
expect consistency with the posterior error; however, the true error, P t ,islessthan
P because ensemble members were selected that did not violate weak nonlinearity.
This created a selection bias that decreases the true initial error of the ensemble. For
all of the cases performed, the P t averages 30 % less than the specified value, P ,
regardless of the method.
With this in mind, the ratio of the posterior error, P a ,tothe prior , P p ,iscom-
pared. The inner-product of the sampled observation locations should be equivalent
to the initial error projected into data-space and the P a is highly dependent upon
the number of observations as expected by ( 10.14 ). As shown in Fig. 10.6 , both
methods tend to underestimate the actual background error. Interestingly, the ratio
depends only upon the choice of R .As R decreases, both assimilation methods
grossly underestimate the error in the background. As R increases, the overfit
method significantly overestimates the error in the background as it relies solely
on the observations, while the constrained method becomes more consistent. The
results are from the regular case, but are consistent with the transition case.
Likewise, ( 10.15 ) is used to compute the posterior R a for comparison with the
prescribed prior R . Figure 10.7 shows that no matter the number of observations
or selection of P , the overfit method always underestimates the actual error in