Localisation, Balance and Choice of Analysis Variable in an Ensemble Kalman
Filter
Jeffrey D. Kepert
Bureau of Meteorology Research Centre, Melbourne, Australia (J.Kepert@bom.gov.au)
Abstract
Theory and practise have shown that an
atmospheric EnKF can produce analyses that are
less well balanced, than other assimilation
methods. Here, the imbalance due to localisation is
explored, and an improved algorithm presented.
The arguments in favour of localisation seem
compelling, yet by weakening the state-dependent
dynamical balances, one of the fundamental
advantages of the EnKF is reduced. Here, the
impact of localisation on balance in an EnKF is
explored and an improved algorithm presented.
2. Idealised studies
1. Introduction
The Ensemble Kalman Filter (EnKF) is a
promising alternative to the Extended Kalman
Filter and 4DVAR for atmospheric and oceanic
assimilation, since it avoids the need for tangentlinear and adjoint models and the need to deal with
a covariance matrix of rank O(106). One of the
main disadvantages is that the EnKF forces the
state covariance to be of excessively low rank: the
ensemble size and covariance matrix rank are in
practice presently of O(102), which is much too
small. This leads to spurious long-range
correlations (which may also be regarded as due
to the true correlation at long range being small
relative to sampling error in the finite ensemble),
and results in the analysis having a poor fit to the
observations. The spurious long-range correlations
cause an incorrect long-range response to a single
observation in both the ensemble mean, and in the
variance, which is reduced globally when a single
observation is assimilated.
To obtain reasonable performance from an
EnKF, it is necessary to filter out these spurious
correlations; this step is known as “localisation”. A
typical localisation approach is to Schur-multiply
the ensemble-estimated covariance matrix by a
compactly-supported
distance-dependent
covariance function (Houtekamer and Mitchell
2001). The function of Gaspari and Cohn (1999, eq
4.10; henceforth GC) has been widely used for
this. This procedure removes all long-range
correlations and increases the rank of the
covariance matrix, thereby improving the ability of
the analysis to fit observations well. However, the
balance information in the covariance matrix is
defined by its null space, so this may generate
imbalance in the analysis (Lorenc 2003).
Houtekamer and Mitchell (2005) show that their
global atmospheric EnKF, with a GC-localisation,
produces analyses with a significant degree of
imbalance, and conclude that this has a
deleterious effect on the system’s performance.
The weakening of balance presents a
potential barrier to operational use of the EnKF.
Here, we show the effect of covariance
localisation on analysis response using the
standard optimum interpolation (OI) covariance
formulation (Lorenc 1981, Daley 1987). The OI
model represents the main balances expected in
the atmosphere at the synoptic and larger scales:
that the analysis increments should be nearly
nondivergent and nearly geostrophically balanced.
Such balances would be expected in long-term
mean covariances in an EnKF, so the impact of
localisation on analyses with the OI model is
indicative of the consequences of localisation
within an EnKF.
2.1 Infinitely dense observations
We modify an example of Daley (1991,
section 5.4) to include the effects of localisation.
Take an infinite one-dimensional domain with
infinitely dense observations of geopotential  and
meridional wind component v, and analyse these
variables. The background errors for both variables
are taken to be 1, while the observation error
densities for both variables are 0.1, as in Daley.
The filtering properties of the analysis under the
above covariance model with geostrophic coupling
parameter μ=1, divergence parameter ν=0, and
various localisations, are shown in Fig 1. The blue
curve is without localisation, while the red GClocalises to 0 at 6L. The -from- response is
broadened and increases at moderate-to-small
scales, because multiplication of the covariance by
the localising function has the effect of reducing
the length-scale. The geostrophic coupling cases,
-from-v and v-from-, when localised retain their
zero response at wavenumber 0, but the peak
response is reduced and shifted to smaller scales,
so the analysis will impose geostrophic balance
more weakly at moderate-to-large scales. The vfrom-v response is unchanged at small scales, but
under localisation has a near-constant response at
large scales. The decrease in response towards
wavenumber 0 shows that the analysis filters
large-scale divergence. The change in this region
under localisation shows that this filtering has been
greatly reduced.
Figure 1 also contains similar curves for an
analysis localised by domain decomposition. Here,
the domains are 2L wide, where L is the
rotational error length scale, and the data selection
halo for each domain extends 6L from the domain
edge. The support of this localisation is thus the
same as the GC-localisation. The deleterious
effects of this abrupt localisation are clearly
considerably less than those of GC-localisation.
Localisation may be regarded as the application of
a high-pass filter to the covariances, which
removes large-scale information. The domaindecomposition localisation corresponds to a filter
with a much less sharp cut-off than GClocalisation, which allows correspondingly more of
the large-scale balance information in the
background covariance to survive. However,
domain decomposition is undesirable in practisein
an EnKF as it introduces noise into the analysis
(Houtekamer and Mitchell 2001).
Figure 1. Bivariate wind-height analysis with infinitely
dense observations, showing the impact of localisation.
Blue: unlocalised. Green: domain decomposition (nearly
obscured by blue). Red: GC localisation.
The response to a single wind observation is
shown in Fig 3. The wind response, of a pair of
gyres either side of the observation, with the
convergence ahead of and divergence behind, are
clearly apparent. Under localisation, the divergent
component is strengthened and so the wind
analysis is similar to that obtained by increasing ν
to 0.18 (not shown). In the unlocalised analysis,
the height response is such that the analysed
winds are everywhere weakly subgeostrophic.
When localised, the distribution of the ageostrophic
flow component changes markedly, and does not
correspond to any other combination of (μ, ν) for
an unlocalised analysis.
Thus part of the effect of localisation on an
analysis using this covariance model is in some
respects similar to reducing μ and increasing ν;
that is, adjusting the balance parameters in the OI
covariance model to a less-balanced setting. The
equivalent changes in μ and ν are larger for shorter
localisation length scales (Fig 4). Recalling typical
mid-latitude values of μ ~ 0.9, ν ~ 0.1, one might
perhaps regard changes of at most 0.05 as
acceptable, equivalent to a localisation half-width
of at least 10L.
Figure 2. Multivariate response to a single height
observation, as described in the text. Top: not localised.
Bottom: GC-localised (to 0 at 12L). Left: Height
analysis. Middle: Wind analysis. Right: Ageostrophic
wind (vectors multiplied by 10).
2.2 A single observation
The two-dimensional analysis response to a
single height observation 1 unit greater than the
background consists of a bell-shaped height
increment and circular flow about it (Fig 2). Here,
μ=0.9 and ν=0.1, so the analysed flow is weakly
subgeostrophic. When the covariance matrices are
GC-localised with zero-radius 12L, the analysed
height field is hardly changed, but the wind is
weakened, so the flow is more subgeostrophic.
The analysis is similar to that found by reducing μ
to 0.82 (not shown).
Figure 3. Multivariate response to a single wind
observation, as described in the text. Top: not localised.
Bottom: GC-localised (to 0 at 6L). Left: Wind analysis.
Middle: Vorticity. Right: Divergence.
Figure 4. The equivalent geostrophy and divergence
parameters in OI covariance model, for various
localisation lengths (in units of L). Base analysis has μ
=1, ν=0. Blue curve: single wind observation. Red curve:
single height observation.
2.3 Discussion
It seems convenient that the imbalances
occur at larger scales, because these scales are
affected by many observations, so spurious
divergence due to observation random error will
cancel out. However, geographically varying
observation bias (due to, say, different satellites)
will be preserved to a much greater degree in a
GC-localised analysis.
The localisation scale necessary to make
the effect “small”, here arbitrarily taken to be
changing μ and ν by 0.05, is rather large at ~10L.
If we take L ~ 500 km, this compares
unfavourably with the 2800 km used by
Houtekamer and Mitchell (2001) and is consistent
with their finding that their system’s analyses are
insufficiently balanced (Houtekamer and Mitchell
2005), although care is needed in extrapolating
this single-observation analysis to the manyobservation case. Their localisation scale was
tuned to optimise system performance for a fixed
ensemble size, and so reflects the conflicting
demands of several aspects of the system,
including the need to control sampling error and to
not excessively harm balance. Relaxing the
localisation scale without degrading performance
would presumably also require an increase in
ensemble size.
3. Choice of analysis variable
The covariances in the OI model above have a
markedly anisotropic structure, the details of which
are crucial to producing balanced analyses. The
weakening of the balance by GC-localisation
demonstrated above occurs because the
localisation does not preserve the fine details of
these structures. For instance, in the wind-wind
correlations, the cross-stream negative lobes, the
along-stream variation in analysed wind speed,
and the along-stream turning of the wind into the
lateral gyres, are more affected by localisation than
the structure near the observation, so the analysis
contains excess divergence along-stream, and
insufficient vorticity cross-stream. Since the root of
the problem lies in the anisotropic structure and
marked cross-correlations present in (u, v)-space,
it is reasonable to explore a transformation of the
analysis variable that eliminates these properties.
Clearly a candidate for such a transformation
would be to perform the analysis in (, )-space,
as the OI covariance model is derived from an
independent isotropic covariance model for (, ).
This transformation is well understood, being used
in operational 3- and 4-dimensional variational
schemes to model the background term and
improve the conditioning of the descent algorithm
(Parrish and Derber 1992). Here we explore the
effect of making similar transformations to the state
vector in an EnKF.
These experiments use the global spectral
shallow-water model of Bourke (1972), truncated
at T31, using an equivalent depth of 1.5 km, with
the addition of a nonlinear normal-modes
initialisation scheme, and weak fourth-order
diffusion with coefficient 2x1014 m4s-1. For
convenience we analyse onto the model Gaussian
grid. Observation locations are chosen from the list
of global radiosonde stations, relocated to the
nearest grid point with duplicates removed, then
reduced by half, leaving 145 points which observe
geopotential and wind every 12 hours, with errors
of 100 m and 5 m s-1 respectively. The truth run
commenced from the ERA-40 500 hPa analysis
valid at 1200UTC on 18 January, 1962. Statistics
presented are for the last 10 days of a 60-day run.
The ensemble size is 64, the GC-localisation goes
to 0 at 6000 km, and covariance inflation of 4% is
applied, chosen to give a flat rank histogram. The
standard perturbed-observation single EnKF is
used.
We consider two variants of this EnKF, one
in which the analysis state vector contains the
conventional variables (EnKF-uv), and the new
variant in which it contains  and  (EnKF-).
Table 1 contains statistics on the relative
performance of the two systems. It is apparent that
the background fields for the EnKF- are
considerably more accurate in  and  than for the
EnKF-uv, and about the same in . The RMS
time tendencies for the forecast step are smaller in
the EnKF-, showing that this is producing
analyses which generate less gravity wave activity
in the model, and are therefore better balanced. A
final diagnostic of balance is provided by the RMS
increment to the analysis produced by the model’s
normal-modes initialisation scheme; these are
generally smaller for the EnKF-, showing that
the analyses are better balanced (the initialised
analyses are here used only for diagnostic
purposes, not as part of the assimilation cycle).
The excess gravity wave activity can also be
seen in the ensemble spread. Fig 5 shows the
mean magnitude of the background  variance in
spectral space, together with the change in
variance under the analysis. The EnKF-uv has
greater background spread, and also substantial
parts of the spectrum where the analysis increases
the variance (blue-green). This is consistent with
the excess gravity-wave activity in this
configuration.
Similar tests were done with different
localisation scales, ensemble sizes and with the
use of an initialisation step in the assimilation
cycle. The conclusions are robust: in all cases the
EnKF- produces better-balanced, more
accurate analyses than the EnKF-uv. Additionally,
the penalty due to localisation was found to be
greater for smaller localisation radii.
BG RMSE 
BG RMSE  (s-1)
BG RMSE  (s-1)
RMS d/dt (m2s-3)
RMS d/dt (s-2)
RMS d/dt (s-2)
RMS a-nmi (m2s-2)
RMSa-nmi (s-1)
RMS a-nmi (s-1)
(m2s-2)
EnKF-uv
327
4.10 x106
1.32 x106
57
4.68 x104
6.25 x104
162
1.25 x106
1.93 x106
EnKF-
297
4.14 x106
1.08 x106
46
4.75 x104
5.13 x104
114
1.28 x106
1.57 x106
Table 1. Statistics for the two EnKF systems, averaged
over the last 10 days of a 60-day run. First three rows:
root-mean-square background error for geopotential,
streamfunction and velocity potential. Next three rows:
Root-mean-square tendencies from the forecast step.
Last three rows: Root-mean-square differences between
analysis and initialised analysis.
Figure 5. Ensemble background variance of  in
spectral space for a single analysis. Top: EnKF-
Bottom: EnKF-uv. Left: Logarithm base-10 of the
ensemble variance in spherical harmonic space. Right:
Change in the logarithm of the variance under the
analysis. Each panel has the meridional wavenumber
on the y-axis, and the zonal wavenumber on the x-axis.
4. Conclusions and Discussion
Covariance localisation was shown to produce
marked analysis imbalance, consisting of large
scale ageostrophic and divergent flow, in the
traditional OI covariance model. For current
localisation methods, idealised calculations
suggest that the amount of imbalance is large
enough to cause problems in an EnKF.
The root of problem is the anisotropic
structure and nonzero cross-covariances in uvspace, which can be reduced by analysing in space. Similar transformations are fundamental to
3DVAR and 4DVAR assimilation systems (for not
dissimilar reasons), so the necessary code for the
transformations is readily available.
One interpretation of localisation is that it is
a method for eliminating covariances that should
be small but because of sampling error, are not.
Traditionally this has focussed on distantdependent correlations, where our knowledge of
the dynamical system tells us the correlations are
spurious. The approach here raises the possibility
of also eliminating suspected-to-be-spurious intervariable correlations (e.g. between  and ), and
could be further extended by replacing  by its
unbalanced part in the EnKF-.
References
Bourke, W.P., 1972: An efficient, one-level,
primitive-equation spectral model. Mon. Wea.
Rev., 100, 683 – 689.
Daley, Roger, 1985: Analysis of synoptic-scale
divergence by a statistical interpolation
procedure. Mon. Wea. Rev., 113, 1066-1079.
Daley, R., 1991: Atmospheric Data Analysis.
Cambridge University Press, Cambridge. 457
pp.
Gaspari, G., and S. E. Cohn, 1999: Construction of
correlation functions in two and three
dimensions. Quart. J. Roy. Meteor. Soc., 125,
723–757.
Houtekamer, P.L. and H.L. Mitchell, 2001: A
sequential Ensemble Kalman Filter for
atmospheric data assimilation. Mon. Wea. Rev.,
129, 123 - 137.
Houtekamer, P.L. and H.L Mitchell, 2005:
Ensemble Kalman Filtering. Quart. J. R.
Meteorol. Soc., in press.
Lorenc, A.C., 1981: A global three-dimensional
multivariate statistical interpolation scheme.
Mon. Wea. Rev., 109, 701-721.
Lorenc, A.C., 2003: The potential of the ensemble
Kalman filter for NWP—a comparison with
4D-Var. Quart. J. Roy. Meteorol. Soc., 129,
3183 - 3203.
Parrish, D.F. and J.C. Derber, 1992: The National
Meteorological Center’s spectral statisticalinterpolation analysis system. Mon. Wea. Rev.,
120, 1747 – 1763.