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BODAS – Dr Peter R. Oke Summary ●
Description ● Characteristics of
BODAS link to a complete list of
publications Oke,
P. R., G. B. Brassington, D. A. Griffin and A. Schiller, 2007: The Bluelink Ocean Data Assimilation System (BODAS), Ocean Modelling, doi:10.1016/j.ocemod.2007.11.002. Oke,
P. R., A. Schiller, G. A. Griffin, G. B. Brassington 2005: Ensemble data
assimilation for an eddy-resolving ocean model. Quarterly Journal of the
Royal Meteorological Society, 131,
3301-3311. The primary test-bed for BODAS in BRAN. BODAS is an ensemble optimal
interpolation system that uses an ensemble of intraseasonal
anomalies from a free running model to estimate the background error covariances (BECs). The
ensemble-based BECs are multivariate and
inhomogeneous and are shown to reflect the length-scales, the anisotropy and
the covariability of mesoscale oceanic processes.
Analyses
of sea-level η, T, S, and horizontal currents (u,v), are computed by solving
the analysis equations,
where
is
the state vector; superscripts a, b, o and * denote analysis, background,
observed and matrix transpose, respectively (here the background field refers
to a model-generated estimate of the ocean state at the analysis time;
sometimes called the first guess); K
is the gain matrix; H
is an operator that interpolates from the model grid to observation
locations; ρ is a correlation function used to localise the
ensemble-based BECs in P; R is the observation error
covariance matrix; and the open circles denote a Schur,
or Hadamard, product (an element-by-element matrix
multiplication). This formulation of the analysis equations is the same as
that of Houtekamer and Mitchell
(2001). Estimates of the BECs in (2) are given by
where n is
the ensemble size and A
is an ensemble of model anomalies. Each anomaly field consists of all model variables
included in (3). The calculation of the anomalies for an EnOI system is critical to the performance of the scheme.
They should be computed in such a way that the scales of variability and
features represented by the anomalies resemble the dominant errors of the
model. For example, for application to OFAM, where we seek to correctly
reproduce the mesoscale variability around Australia, we expect that the
errors of an individual forecast will be dominated by the errors associated
with mis-placement of eddies. Section 4 shows that this turns out to be true, although
it is also true that errors at other scales exist. We hope to address this
compromise in the future. Consequently, for application to OFAM, the ensemble
of anomalies are generated by calculating intraseasonal
anomalies derived from the spin-up run:
where α is a scalar that can tune the
magnitude of the covariances for a particular
application (for BRAN1.5, α=1); The BECs
in (4) are localised in the horizontal
around each observation in (2) using the localising correlation
function ρ. Elements of ρ are defined by the
quasi-Gaussian function of Gaspari and Cohn (1992),
after Houtekamer and Mitchell (2001). Localisation has been shown to
reduce the effects of sampling error for applications of an EnKF (e.g., Hamill et al., 2001) and EnOI
(Oke et al., 2007). The localising correlation
function in ρ forces the BECs to reduce
to exactly zero, over L° from an observation location. Note that in
the application, we do not localise the covariances
in the vertical direction. The present implementation of BODAS uses a uniform
horizontal radial distance L = 8°, corresponding to an e-folding
scale of about 3.5°. This has several ramifications for the system’s
performance. Firstly, the rank of the estimated BECs
in P is increased
significantly. Using an ensemble size of n, the rank of P without localisation is at most
n-1. By contrast, using an ensemble of
only 72 with localisation with L = 8° we estimate that the
effective rank of ρ The primary
test-bed for BODAS is BRAN. BRAN is a multi-year integration of OFAM, where BODAS
is used to sequentially assimilate observations once every 7 days.
Observations assimilated in BRAN include As stated above, one of the advantages of EnOI is that the BECs are
inhomogeneous and anisotropic, reflecting the variability and length-scales
of the ocean circulation. An example of the anisotropy and inhomogeneity of the ensemble-based BECs
used by BODAS is shown in Fig. 4. The chosen examples show the localised
ensemble-based cross-correlation between sea-level at a reference location
and sea-level in the surrounding region. These fields demonstrate the region
of influence of an observation at the reference location. Where the
correlation is positive, the increment that is given by the term K(wo-Hwb) in (1), due to each observation has the same sign as the
background innovation (wo-Hwb). So if the observed sea-level is higher than the
background field at the reference location, in the absence of other
observations in the same region, the solution to (1) will produce an increment that is also positive.
The magnitude of the increment depends on the relative magnitudes of the
estimated background error and observation error covariances
(i.e., the relative magnitudes of H(ρ Fig.
4. Examples of the ensemble-based cross-correlations between sea-level
at a reference location, denoted by the star, and sea-level in the surrounding
region for a reference location (a) on the continental shelf, (b) over the
continental slope and (c) over the deep ocean off eastern The reference
locations for the examples presented in Fig.
4 are at 32.5°S, corresponding to the typical separation point of the EAC
(Godfrey
et al., 1980). These examples include correlations when the reference
location is on the continental shelf at 115 m depth, where the
correlation field has short decorrelation scales in
the across-shore direction and long decorrelation
scales in the alongshore direction (Fig.
4a). The long length-scales in the alongshore direction are probably a
reflection of the covariability associated with
northward propagation of slow-moving coastal trapped waves and the
along-shore advection of the EAC and wind-driven circulation. Also shown is a
correlation map when the reference location is over the continental slope at
1800 m depth (Fig.
4b), less than 50 km east of the shelf example (Fig.
4a). This field shows a more isotropic correlation, but with a tendency
to have higher correlations to the east and south-east, in the typical
direction of the EAC as it separates from the coast. The final example
considered here shows correlations when the reference location is over the
deep ocean at 4500 m depth (Fig.
4c). This field shows a somewhat isotropic correlation field, but also
shows areas of weak negative correlation to the north-east and south-west.
These negative correlations may be related to variations in sea-level
associated with the typical eddy field in this region. The correlation fields
presented in Fig.
4 highlight the anisotropy and inhomogeneity of
the ensemble-based correlations, with relatively long quasi-isotropic
correlations in the deep ocean, transitioning to very anisotropic
correlations nearer the coast. An example of
the multivariate nature of BODAS is demonstrated in Fig.
5 and Fig.
6, showing the increments from a single observation analysis (i.e., where
an analysis is computed by combining a single observation with a background
field). In this example, an observation of sea-level at the coast is presumed
to be 20 cm lower than the background sea-level. BODAS is used to
calculate the increments to the full model state (η,T,S,u,v)
in the surrounding region. We choose to use sea-level from a coastal location
off Fig.
5. Increments to sea-level (grey scale) and surface currents (vectors)
based on a single observation of sea-level, denoted by the Fig.
6. Increments to (a) sea-level, (b) across-shore currents (dashed
contours are offshore; bold contour is zero) and (c) along-shore currents
(solid contours are into the page; bold contour is zero); and the background
field (solid contours) and analysis (dashed contours) for (d) T and
(e) S, based on a single observation of sea-level at the coast as in Fig.
5. Decreased
sea-level at the coast may be due to a number of factors that include, but
are not limited to, wind-driven upwelling. However, the dominance of locally
wind-driven circulation in this region leads us to the expectation that it
will also dominate the statistical properties of the ensemble fields here. We
find that the increments to sea-level and surface currents in Fig.
5 are consistent with wind-driven upwelling, with reduced sea-level along
the coast, a relatively strong coastal jet and weak south-westerly flow in
the deep ocean. Fig.
6 shows the impact of the observation along a shore-normal section
offshore of the observation location. This figure shows that the negative
sea-level increment is strongest at the coast as we expect, and approaches
zero moving offshore in a quasi-exponential fashion. Increments to the
across-shore currents show an offshore flow of up to 5 cm s−1
over the top 30 m, that is consistent with an offshore wind-driven Ekman layer, and a weak, shoreward return flow at depth.
Increments to the along-shore currents are consistent with a baroclinic, wind-driven coastal jet, with the strongest currents
of 0.5 m s−1 at the surface near the coast. Fig.
6d and e shows the background and analysed T and S fields.
Both of these fields show an uplift of isotherms and isohalines, with an
implied vertical excursion of about 25 m near the shelf break and
40–50 m over the upper slope. From this analysis, we conclude that
the statistical properties of the ensemble are consistent with a conceptual
model of wind-driven upwelling in the region considered. The analysis
described above indicates that if the model does not produce an upwelling
event, due to incorrect surface forcing for example, and we assimilate a
single observation of sea-level at the coast that reflects the upwelling
through reduced sea-level, then BODAS will compute increments that more
closely match the observation at the coast, and in a manner that is
consistent with wind-driven upwelling. While this feature is presented here
as a benefit of EnOI, there is also a down-side.
Suppose the model-observation mismatch is due to a mis-represented
process other than wind-driven upwelling. Say, the encroachment of an eddy,
or the propagation of a coastal trapped wave. Then the EnOI-derived
increments will still be consistent with upwelling as in Fig.
5 and Fig.
6. This may not be desirable, and may result in a reanalysed state that
is somewhat inconsistent with reality. However, we note that the example
described here is very idealised. In practice, we
typically assimilate multiple observations of various types (e.g., sea-level,
T and S) and in the presence of additional observations that
provide a more complete picture of the ocean state, BODAS will compute
increments that more appropriately represent the true circulation. Another
limitation of EnOI that can be seen from the
example described by Fig.
5 and Fig.
6 is that it implies a symmetry in the increments for an
observation-model mis-match of the opposite sign.
Suppose that the observation-model difference considered above is reversed.
That is, the observation is 20 cm higher than the background sea-level.
In this case, the increments simply have the opposite sign to those presented
in Fig.
5 and Fig.
6. This might occur if a modelled upwelling is too strong, or perhaps if
the model fails to represent a downwelling event.
However, we note that studies into the dynamics of idealised wind-driven
upwelling (e.g., Allen
et al., 1995) and wind-driven downwelling
(e.g., Allen
and Newberger, 1996) on the continental shelf
have shown that they are very different, and do not simply result in
symmetric anomalies about some mean field as the EnOI-based
increments would imply. These problems are the result of assuming ergodicity in the BECs. A
better approach is to employ some flavour of the EnKF
(e.g., [Evensen,
2003] and [Sakov
and Oke, in press]), where the anomalies in (5)
evolve in time and more accurately reflect the dominant dynamical processes
for a particular time. However, these types of filters require an ensemble of
states to be evolved, which is currently too computationally expensive for
applications as large as OFAM. |
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Last updated 22/09/06 | Legal Notice and Disclaimer | Copyright |
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