Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 158

as was already discussed in detail at the end of Sect. 7.2 . Note that this result is
rather remarkable in so far as the phase location of the feature is located precisely at
its expected value and one might hope that this would be the location that an EnKF
DA system would perform well.
In fact, an EnKF DA system performs better (e.g., similar to quadratic nonlinear
regression), not when the innovation is very small, but actually when the innovation
is equal to its expected value ( Hodyss 2011 ). This result can be seen in Fig. 7.4 c.
In Fig. 7.4 c the true state is the Gaussian phase error model ( 7.42 )fora ,

i.e. the phase location observation is taken as situated at one standard deviation and
therefore the amplitude of the innovation is essentially its expected value. Again, the
particle filter's estimate of the posterior mean is indistinguishable from the true state
in this case. However, the EnKF result is now basically identical to the Quadratic
Ensemble Filter result. Mathematically, we can understand that this occurs when the
quantity in square brackets in ( 7.27 ) vanishes, and this happens for innovations that
have the amplitude of their expected value

' D

. h v 2 i 2 /

because the third moment in
observation space (location observations) is zero for Gaussian phase errors.
In Fig. 7.4 e we show an example for which the true state is the Gaussian phase
error model ( 7.42 )fora , i.e. the phase location observation is taken as

situated at two standard deviations and therefore the amplitude of the innovation is
significantly larger than its expected value. Again, the particle filter's estimate of the
posterior mean is indistinguishable from the true state in this figure. However, in this
case the EnKF's estimate of the posterior mean and that of the Quadratic Ensemble
Filter is now quite different. When the observation errors are low but the innovation
is larger than its expected value a common way in which the EnKF is in error is to
produce an estimate of the posterior mean that has too much amplitude. This can
be seen in Fig. 7.4 e by the large values in the estimate of the posterior mean above
1,000 mb and below 975 mb. Note that the true posterior mean does not go above
1,000 mb or below 975 mb. In addition, the phase location of the estimate from the
EnKF typically does not shift far enough towards the true location. The Quadratic
Ensemble Filter produces an estimate that has significantly reduced issues with the
amplitude of the estimate of the posterior mean and the phase location and this is
obviously due to its ability to account for the nonlinearity seen in Fig. 7.3 .
Next, we will consider the situation where the observation error variance is
substantially larger:

' D 2

. In this case the true posterior mean from the particle
filter will not be particularly close to the true state as the observation is not accurate
enough to distinguish the location of the feature exactly. This can be seen in
Fig. 7.4 b. In this case the innovation is again zero because the observation is taken
to be located at the mean phase location. Again, the estimate of the posterior mean
from the EnKF is just that of the prior mean. In contrast, the Quadratic Ensemble
Filter now produces an estimate of the posterior mean that is very close to that of
the true posterior mean from the particle filter. In Fig. 7.4 d is shown the case where
the location of the observation is at one standard deviation. As one can see from
Fig. 7.4 d the state estimate from the EnKF is now again very close to that of the
Quadratic Ensemble Filter. Therefore, for this size of innovation both linear and
quadratic nonlinear regression give very similar estimates of the posterior mean
and they are quite close to the true posterior mean. In Fig. 7.4 f is shown the case

R D 1