Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 155

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Phase Location

Fig. 7.3 The distribution of pressure at x D 2 as a function of the location of the feature

model ( 7.42 ). Note that the structure of the distribution in this plane focuses all
the members along a single loop structure. This loop arises because for a value of
pressure at, say

x D x 0 , there are always two different phase values, '

,thatare

x D x 1 .We
emphasize that the loop structure is not restricted to just the relationship between
the center and the inflection point, but also occurs between all other locations as
well; the loop structure shown in Fig. 7.2 between other locations differs only in
the shape of the loops. The existence of these loops has important implications to
the accuracy of the DA. Because the relationship between any two locations is not
only nonlinear but also multi-valued the DA must be able to choose which side of
the loop is the correct side. Subsequently, a single observation of pressure cannot
discern the location of the feature because the distribution is always multi-modal.
Moreover,asshowninFig. 7.2 the loop structure of the prior distribution assures
that the prior mean will not be a state representative of any particular member. The
ensemble mean value in this plane is denoted in Fig. 7.2 and can be seen to be
well away from the loop. This loop structure makes clear the complexity of these
distributions and implies significant high-order multivariate moments.
In tropical cyclone data assimilation the location (or phase,

consistent, which leads to two different possible pressure values at

'

) of the minimum
central pressure is an often used observation. Therefore, it is of interest to examine
the prior distribution of phase locations (

'

) plotted against the values of pressure at
some location we might update with that observation of location. In this respect this
tells us the relationship between the state and the prior estimates of the observed
variables. Figure 7.3 shows what this looks like for phase locations (

'

) plotted

x D 2

against the values of pressure at for our Gaussian phase error model ( 7.42 ).

Because the disturbance structure is simply a function of the phase the distribution
here is not a loop like Fig. 7.2 but simply the actual structure of the Gaussian phase
error model ( 7.42 ). Nevertheless, this functional dependence is clearly nonlinear and
as we show next will lead to significant difficulties with Kalman filter-based DA.