Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 164

posterior error variance as a function of the position observation, which is of course
the estimate of the posterior error variance obtained from the particle filter, has
two maxima. One maximum is at the location of the prior mean and the other is
near to the location of the observation. The structure of the true posterior error
variance, in particular its multi-modal character, is again very similar to that found
in the idealized experiment (Fig. 7.5 f), which gives confidence to our estimate of
the true posterior error variance using the limited ensemble size of 80 members. In
Fig. 7.7 b, c are the EnKF and quadratic ensemble filter estimates of the expected
posterior error variance. Note that while the quadratic posterior error variance is
less than that estimated by the EnKF neither has the correct structure of the posterior
error variance; both have a single maximum in the vicinity of the prior mean location
much like that of the idealized experiment in Fig. 7.5 f. The fact that the ensemble
mean from the quadratic ensemble filter is much better than that from the EnKF
but that the ensemble generation is not significantly better underscores the fact that
contemporary ensemble generation techniques do not properly account for the latest
set of observations.

7.5 Summary and Conclusions

This chapter has explored in detail the issues surrounding the impact of phase errors
on the ability of traditional ensemble-based Kalman filtering (EnKF) algorithms
to accurately reproduce the posterior mean and perturbations to that mean that
sample the posterior distribution. We began by illustrating the relationships between
EnKF algorithms and linear and nonlinear regression. Here, we saw that quadratic
nonlinear regression is simply the EnKF with a correction term that provides some
accounting for the prior third moment. The prior third moment turns out to be of
some significance as prior distributions whose uncertainty arises from uncertainty
in the location of the feature have large third moments. This third moment (or
skewness) of the prior distribution was shown to lead to difficulties for EnKF
algorithms.
We have shown that an important issue with the estimation of the posterior mean
from an EnKF algorithm is the size of the innovation. In situations with non-zero
prior third moments, such as phase uncertainty, the posterior distribution is almost
always a curved (nonlinear) function of the innovation ( Hodyss 2011 ). This implies
that for large innovations the EnKF will always produce significant error because
it is a linear function of the innovation. More surprising however is the fact that
the EnKF will also make a significant error in its estimate of the posterior mean
whenever the innovation is very small. This is because whenever the posterior third
moment is large and the innovation is small the posterior mean curves away from
the prior mean. This leads to significant error in the estimate of the posterior mean
because for small innovation the EnKF estimate of the posterior mean is always
the prior mean. In fact, the size of the innovation for which an EnKF algorithm is