Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 144

over more complex methods. Even with its reliance upon only the first two moments
the application of the EnKF in the meteorological community has been met with
considerable success in a wide-range of applications (e.g., Houtekamer et al. 2005 ;
Szunyogh et al. 2008 ; Meng and Zhang 2008 ; Torn and Hakim 2008 ; Whitaker
et al. 2008 ; Anderson et al. 2009 ).
There are, however, several unresolved issues with the application of the EnKF to
the highly nonlinear dynamics inherent to meteorological flows at high resolution.
Situations in which the EnKF is known to have some difficulty, and where
nonlinearity may be significant, include the assimilation of: vortex position ( Lawson
and Hansen 2005 ; Chen and Snyder 2007 ), radar observations ( Dowell et al. 2011 ),
parameter estimation ( Hacker et al. 2011 ), and observations over a long assimilation
window ( Khare et al. 2008 ). We speculate that one reason the assimilation in
these situations is sometimes difficult is the fact that the relationship between the
prior estimates of the observed variables and the state-vector is nonlinear. This
nonlinearity may come about from nonlinearity in the model operator, which is
described through the dynamics of the physical system, or from nonlinearity in
the observation operator used to observe the system. In either case, this nonlinear
relationship leads to skewed (non-zero third moment) posterior distributions that,
as has been discussed by Hodyss ( 2011 ) and reviewed below, results in suboptimal
behavior from the EnKF.
Previous work towards state estimation techniques for nonlinear modeling sys-
tems has been discussed by Kushner ( 1967 ); Jazwinski ( 1998 ; Chap. 9 ), Anderson
and Anderson ( 1999 ), and Julier and Uhlmann ( 1997 , 2004 ). However, in the
geophysical sciences work towards explicitly accounting for the non-Gaussian
aspects of the prior within an ensemble DA framework has emphasized particle
filtering [See van Leeuwen ( 2009 ) for a comprehensive review]. In particle filtering
one estimates the probability that a particular realization is the true state by
comparing against observations. Given these probabilities (sometimes referred to
as weights) one can do such things as make a state estimate based on the mean of
the pdf or randomly sample from the pdf to generate an ensemble consistent with
observational and prior uncertainties. Another way to get non-normal distributions
is through nonlinearity in the observation operators or the non-normality of the
observation likelihood. Two examples of ensemble filters that are aimed at the
situation where the observation operator is nonlinear or the observation likelihood is
non-Gaussian (but the prior distribution is normal) are the work of Zupanski ( 2005 )
and Fletcher and Zupanski ( 2006 ).
The focus here however will be on neither model nor observation operator
induced nonlinearity; here we will focus on the nonlinearity, or more specifically the
non-Gaussianity, that arises from phase errors. This “nonlinearity” that results from
the uncertainty in the location of the feature will be shown to arise from a nonlinear
relationship between the uncertainty in the location of the feature and the state
variables describing the physical system. We show below that prior distributions
whose uncertainty arises from errors in the location of a feature leads to significant
skewness and hence significantly non-Gaussian distributions. The distributions that
arise from phase error uncertainty will be shown to have surprisingly complex