Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 165

most accurate is when the magnitude of the innovation is approximately equal to its
expected value.
We have shown that a significant issue with the generation of an ensemble that
correctly samples the posterior distribution is in the calculation of the posterior
error variance. One way to view ensemble generation is simply as a method to
estimate the posterior error covariance matrix. Because normal distributions have
posterior covariance matrices that are independent of the innovation (observation)
this has led to a significant number of ensemble generation algorithms based on
the expected posterior error covariance matrix. This unfortunately is an assumption
and it fails most strongly whenever the posterior distribution has non-zero third
moments (skewness). The generation of an ensemble in feature-based systems with
observations of location leads to this issue becoming extremely important. We
have seen in both idealized cases and in a real tropical cyclone that the correct
posterior error covariance matrix knows the location of the observation whereas the
expected posterior error covariance matrix does not. This leads to two problems:
(1) the uncertainty is not appropriately centered in the correct location and (2) the
structure of the variances and co-variances, even if they were correctly located
through a shift in their position, are simply not correct. The first problem leads
to the uncertainty being in the wrong location and subsequently not accurately
predicting the probability of particular events. The second problem implies that
the ensemble generation cannot produce structures that are self-consistent with the
structures within the prior ensemble. This means that the TCs that are produced
by the ensemble will not have the correct relationships between variables because
the co-variances are simply incorrect. It is important to realize that this issue was
not corrected by the use of quadratic nonlinear regression. This is because while the
polynomial expansion in the innovation suggested in Sect. 7.2 provides a convergent
estimate of the posterior mean with higher-order approximations, it does not in
fact provide a convergent estimate of the posterior error covariance matrix as a
function of the innovation when combined with the ensemble generation algorithms
referred to as perturbed observations ( Houtekamer and Mitchell 1998 ; Burgers
et al. 1998 ; Evensen 1994 ) or square root filters (Anderson 2001; Bishop et al. 2001 ;
Tippet et al. 2003 ). The reason for this is because these algorithms for generating
ensemble perturbations are based on an estimate of the posterior error covariance
matrix that is incorrect in situations with significant third moments. Both of these
ensemble generation algorithms are based on a covariance matrix that is obtained
by estimating the weighted averaged over all possible posterior covariance matrices
rather than the one associated with the latest innovation.
Nevertheless, from a practical point of view, the fact that the estimate of the
posterior mean from the quadratic ensemble filter is more accurate than that from
an EnKF and subsequently that the posterior error variance is smaller may in fact
translate into a better performing data assimilation system in the presence of phase
uncertainty. Research in this direction with tropical cyclones and other atmospheric
phenomena is ongoing.