Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 151

EnKF may make is simply noted by asking when the new terms in ( 7.27 and
7.28 ) dominate over those of the EnKF; there are two important situations when
this happens. The first situation in which skewness is a significant issue for an
EnKF is when the innovation is very large when compared with its variance. The
EnKF makes a correction to the prior mean that goes linearly with the innovation.
However, depending on the direction of the skewness of the posterior (sign of
the third moment) and the sign of the innovation this correction will either be
too large or too small. This is due to the fact that the true posterior mean is a
curved (nonlinear) function of the innovation whenever the posterior has significant
skewness ( Hodyss 2011 ). Therefore, the EnKF's linear (in the innovation) estimate
of the curved posterior mean will always contain significant error whenever the
innovation and the skewness of the posterior are large.
The second situation illustrated by ( 7.27 and 7.28 ) in which skewness is a
significant issue for an EnKF occurs when the innovation is very small when
compared with its variance. The fact that the EnKF makes significant errors when
the innovation is small is rather surprising and is a result of the fact that when the
innovation is small and the prior third moments are large the f 0 term dominates
the expansion. Note that when the innovation is small the estimate of the posterior
mean from the EnKF (See 7.21 ) is just that of the prior mean. However, when there
is significant skewness in the prior distribution the prior mean is not a good estimate
of the posterior mean. This can be seen in ( 7.7 ) through the importance of the f 0 term
for small innovation. The point to be made here is that whenever there is significant
skewness, and the innovation is very small, the posterior mean and the prior mean
will differ. However, in this situation the EnKF will not make a correction to the
prior mean and in terms of its estimate of the posterior mean it will behave as if
there were no observation to assimilate. More about these issues will be discussed
below in Sect. 7.4

7.3 Distributions Arising from Phase Errors

Imagine a localized disturbance to a fluid, such as a tropical cyclone (TC). Suppose
that this disturbance has a pressure field that appears as in Fig. 7.1 a. Further suppose
that the center (point of lowest pressure) of this disturbance is situated at a grid point
.x D x 0 D 0/

of our model and we are, for now, interested in the moments of the
prior distribution at this grid point. The uncertainty in the prior is assumed to arise
from a normally distributed random variable ' N.0; 2 / denoting the location of

the disturbance. No amplitude (structural) uncertainty will be considered.
The uncertainty in phase has been assumed to be normally distributed in order
to show that non-Gaussianity in a state variable such as pressure can arise even
from normally distributed phase errors. There is however no reason to expect that
phase errors in complex fluid dynamical systems would be normally distributed.
This point is explored further in Appendix 2. In Appendix 2 we show through the
method of characteristics that in general sheared flows the variable translation speed