Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 156

7.4 DA in the Presence of Phase Errors

7.4.1 Idealized Cases

In this section we will perform DA on the distribution ( 7.42 ) studied in Sect. 7.3 .
In particular, we set the distribution shown in Fig. 7.3 to be our prior. This implies
that we will simply observe the location of the feature and attempt to update the
state. As a baseline to compare linear regression (EnKF) against quadratic nonlinear
regression (quadratic ensemble filtering) we will employ a particle filter [See van
Leeuwen 2009 ) for a review]. Because we will be using an ensemble of size 20,000
in this section the posterior mean obtained from the particle filter will be for all
intents and purposes identical to that of the un-approximated application of Bayes'
rule. Because we have one observation (the location of the feature) and 20,000
ensemble members in this particle filter these experiments will not be contaminated
by the detrimental issues of limited ensemble size discussed in Snyder et al. ( 2008 ).
In any event, the goal in these experiments will be to understand when linear
and/or quadratic nonlinear regression can and cannot get close to the result of the
un-approximated application of Bayes' rule.

7.4.1.1 Estimating the Posterior Mean

To begin we will examine the situation of observing the location of the feature
with very low observation error variance, . When the observation error

is very low an EnKF based DA system will make substantial corrections for large
innovations. However, because the relationship between the observation and the
state space is nonlinear (See Fig. 7.3 ) the EnKF correction will not be accurate for
all values of the innovation. We proceed to illustrate this in detail next.
In Fig. 7.4 a, c, e is shown the results of the experiments with low observation
errors. In each figure is shown the true state that is observed. In Fig. 7.4 a the true
state is the Gaussian phase error model ( 7.42 )fora

R D 0:01

' D 0 Because the distribution

of phase errors is , the prior mean is identically zero and because the

observation in this case is zero the innovation is also identically zero. For this
innovation the particle filter's estimate of the posterior mean is for all intents and
purposes identical to the true state and hence indistinguishable from the true state in
this figure. What this means is that a high quality observation of the location of the
feature should identify that feature precisely because there is no uncertainty in the
structure of the feature. Recall that for each value of location (the observation in
this case) the distribution in Fig. 7.3 is a nonlinear but single-valued function. This
implies that if one's DA system can handle the nonlinearity, then one observation
can identify the feature quite closely. However, in Fig. 7.4 a one can see the obvious
result of the zero innovation being that the EnKF estimate of the posterior mean is
actually identical to the prior mean, i.e. no correction has been made . In contrast,
the Quadratic Ensemble Filter makes a correction even when the innovation is zero,

' N.0;2/