Dual Kalman Filters for Simultaneous Assimilation of Physical and
Biochemical Data into a Marine Ecosystem Model
George Triantafyllou
Hellenic Center for Marine Research, PO BOX 712, 19013, Anavyssos, Greece
Email: gt@ath.hcmr.gr
Ibrahim Hoteit
Scripps Institution of Oceanography, La Jolla, CA 92093-0230, USA
Email: ihoteit@ucsd.edu
Gerasimos Korres
Hellenic Center for Marine Research, PO BOX 712, 19013, Anavyssos, Greece
Email: gkorres@ath.hcmr.gr
Abstract: We present a reduced dual Kalman filter approach to simultaneously assimilate physical and
biochemical data into a complex three-dimensional ecosystem model of the Eastern Mediterranean. The
ecosystem model is composed of two on-line coupled sub-models: the Princeton Ocean Model (POM) and
the European Regional Seas Ecosystem Model (ERSEM). In the dual approach two Kalman filters acting
independently on the physics and the ecology are considered to assimilate available data to each subsystem. Here we used the Singular Evolutive Extended Kalman (SEEK) filter which operates with low-rank
error covariance matrices to reduce the heavily computational burden of the extended Kalman filter.
Results of preliminary twin experiments are presented and discussed.
1.
Introduction
Marine ecosystem modeling requires the
coupling of two complex models: the physical
model that describes the currents of the modeled
area, and the biochemical model that describes the
interactions between the different ecological
species. Up today, most studies only considered
the assimilation problem with one of the two
models while assuming that the other one is
perfect. However, assimilating data into one
system only may result in misalignments of the
physical and biological fronts, giving rise to
spurious cross-frontal fluxes of biological
quantities. For instance, assimilation of biological
data alone often leads to spurious ecological
responses (e.g. enhanced productivity). Likewise a
perfect model assumption is far too optimistic and
obtaining reliable estimates of the ecology, for
example, using imperfect physical forcing can be
very difficult. It is therefore necessary to constrain
both models simultaneously with physical and
biological observations, to improve their behavior
and to assure consistency between their respective
analyses. In other words, a successful ecosystem
assimilation system requires the coupling of a
biological and a hydrodynamical assimilation
system capable of producing relevant physical
fields, supportive of the newly analyzed biology.
In the contest of Kalman filtering, this
problem can be addressed following ‘’the Joint
approach’’ or ‘’the dual approach’’ (Wan and
Nelson, 2000). Both approaches were originally
designed for the estimation of the model state
concurrently with the model parameters using an
analogous filter. We generalize them here to the
problem of estimating the state of the ecological
model concurrently with the physical forcing which
evolves in time according to a dynamical model.
Another potential difference in our case is that
ecological as well as physical data can be
available for assimilation.
The joint approach is the simplest among
the two to conceptualize: the physical state vector
is simply appended into the ecological state
vector, to form a single state vector for the
coupled system. The physical and ecological
observations are also appended together into one
observation vector. The time update for the
ecological part and the physical part is performed
by each model, but the entire augmented
covariance matrix is propagated as one. The dual
filtering approach adopted in this study,
intertwines a pair of distinct Kalman filters; one
estimating the ecology and the other estimating
the physics. In this sense, the dual approach
respects more the one-way coupling nature of our
ecological model by allowing the filter to correct
the ecology independently from the physics,
Food Web
than that of different species) where organisms
wi t h s i m i l ar p r o p e r t i es a r e
Pelagic
Model
Irradiation
Wind
Heat
Flux
PicoPhyto
NanoPhyto
POC
Flagellat
es
S
i N
PO
O3 N
H
Diatom
s
4
Bacte
ria
4
C
O
DOC
2
Mesozoo
Carnivorous
Microzoo
o
C
Heterotr.
Flagellates
N, P, Si
Physical
Model
Mesozoo
Omnivorous
Sedimentation
Benthic
Fig. 1. A schematic description of theModel
ecosystem
functional groups and their trophic relations.
resulting in more degrees of freedom to better fit
the data. Another advantage of the dual approach
is that it allows applying different degrees of
simplification to each filter according to the needs
of the system and the user. This was of crucial
importance for the realization of the present study
as it enabled to significantly reduce heavily
computational load associated with the application
of two advanced Kalman filters with two state-ofthe-art ecological and ocean circulation models.
Hereafter we briefly describe the ecosystem
model and the assimilation method before
presenting the results of the assimilation
experiments.
2.
The Ecosystem Model
The ecosystem model consists of two, online coupled, sub-models: the Princeton Ocean
Model (POM) (Blumberg and Mellor, 1987), which
describes the hydrodynamics of the area, and
provides the physical forcing to the second submodel, the European Regional Seas Ecosystem
Model (ERSEM) ( Bar etta et al., 1995), as
summarized in Fig.1. POM is a three dimensional
time dependent primitive equations ocean model;
the equations are solved over an Arakawa-C
d i f f er e n c i n g s c h em e a n d a  - c o o r d i n a t e s
discretization in the vertical. Time integration is
achieved through an explicit scheme in which the
barotropic and baroclinic modes are integrated
separately using a leap frog scheme with different
time steps. ERSEM describes the biogeochemical
cycles and has been successfully applied in a wide
variety of regimes from coastal eutrophic to open
sea oligotrophic systems and on a variety of spatial
scales. The use of a functional group idea (rather
Fig. 2. Model domain.
grouped together, increases ERSEM portability to
any area. The biotic system encompasses the
three major types (producers, consumers and
decomposers) with each type being further
subdivided, increasing the required complexity
into 88 state variables.
The modeled area covers the entire eastern
Mediterranean (Fig.2). A horizontal resolution of 6
minutes was chosen producing a 165106
horizontal grid points. The -layers were
distributed logarithmically near the surface and the
bottom in order to better resolve the surface and
the bottom boundary layers respectively. The
model was
initialized
with
climatological
objectively analyzed temperature and salinity
profiles from the Mediterranean Ocean Database
(MODB-MED4). Initial velocities were set to zero.
Wind stress fields were derived from the ECMWF
6-hour reanalysis data. Biological variables were
initialized through a 1D ecological model as
described by (Triantafyllou et al., 2004).
3.
The Assimilation Method
The assimilation scheme is based on the
Singular Evolutive Extended Kalman (SEEK) filter
which has been developed by Pham et al. (1997)
as a reduced-rank Extended Kalman (EK) filter
with application to highly dimensional systems ( n )
in mind. It basically reduces the prohibitive
computational burden of the (EK) filter associated
with the huge size of the filter’s error covariance
matrices P by operating with low-rank ( r ) error
T
covariance matrices, i.e. P  LUL , where L and
U are n  r and r  r matrices. Under this
assumption, the algorithm of the EK filter remains
mostly unchanged. Only the evolution of P is
avoided and replaced by those of L and U . The
EK filter correction is then only applied along the
directions of L , which we refer to as the “correction
basis” of the filter.
At the initial time, the correction basis is
initialized through an Empirical Orthogonal
Functions (EOF) analysis, which is generally
applied on a historical set of model outputs. The
evolution of L is then performed with the tangent
linear model. This can be numerically very
demanding, as it requires r  1 model integrations
to linearize the model around every column of L .
Several studies demonstrated however that the
evolution of L could be omitted for weakly variable
models (Hoteit et al., 2002) without significantly
affecting the filter’s performance. The resulting
Singular Fixed Extended Kalman (SFEK) filter,
which makes use of a set of EOFs as invariant
correction basis, can be numerically r  1 times
faster than the SEEK filter.
In separate recent studies, the authors
noticed that the SFEK filter behaved fairly well
when applied to ERSEM (Triantafyllou et al.,
2004), while the evolution of L was needed for the
assimilation with POM (Korres at al., 2005). These
findings were very beneficial for setting up a
computationally reasonable dual filtering system in
which the assimilation system ERSEM/SFEK was
coupled with the system POM/SEEK allowing for a
significant reduction in computing time with respect
to the joint approach, since the latter makes use of
only one single state vector, and the
implementation of only one filter (SEEK or SFEK).
However, the use of the SEEK filter was necessary
to obtain satisfactory performances with POM.
It deservers noting that in its general form,
ecological data should be also assimilated into the
physical POM/SEEK system. However this was not
considered in the present study to reduce heavy
computational load required for the linearization of
the biological model with respect to the physics.
4.
Experiments and Discussion
The effectiveness of the dual assimilation
ecosystem system was evaluated following a
‘’twin-experiments'' approach in which the ‘’truth'' is
assumed to be provided by the model itself. Twin
experiments allow assessing the filters behavior on
non-observed
variables,
as
all
uncertain
parameters are known by design. The model
statistics were also used for the initialization of the
filters.
After a four years integration of the
ecosystem model to achieve a quasi adjustment of
the model dynamics, another integration of two
years was carried out to generate two historical
sequences  and  of 365 POM and ERSEM
state vectors. The states were sampled every two
days. The ‘’physical filter’’ and the ‘’ecological
filter’’ were then respectively initialized by the
means of  and  . Reduced-rank
approximations of the filters initial error covariance
Physics
Ecology
Model
POM
ERSEM
Filter
SEEK
SFEK
Data
SSH
CHL
Filter Rank
50
20
Table 1. Table summarizing the characteristics of each
component of the dual assimilation system.
matrices were obtained by applying separate EOF
analysis on  and  . Prior to the analysis, the
models variables were normalized by the inverse
of the square-root of their domain-averaged
variances to make the distance between different
model state variables independent from unit of
measure. The ranks of the physical filter and the
ecological filter were set to 50 and 20, as the first
50 physical EOFs and 20 ecological EOFs explain
more than 80% and 90% of the systems total
variance, respectively.
For the twin-experiments, pseudoobservations of sea surface height (SSH) and
chlorophyll (CHL) data were assumed to be
available every two-days over the whole surface
of the model domain. These observations were
extracted from a set of 45 reference physical and
ecological states simulated by the coupled model
over a three months period (from March 5th to
June 5th). Random Gaussian noises were also
added to the pseudo-observations in order to build
a more realistic framework. The reference states
were afterwards used to evaluate the filters
estimates, relative to the model free-run
estimates. The free-run refers to model integration
(without any assimilation) during the assimilation
period and initialized from the filters initial
conditions. The quality of the filters’ estimates was
measured for all physical and ecological state
variables by the relative error (RMS), which is
defined as the ratio between the filter/reference
and free-run/reference domain misfits. Relative
errors smaller than unity indicate that the solution
of the assimilation system is closer to the truth
than that of the model free-run.
Assimilation experiments were performed
to assess the performance of the filters. Fig.3 and
Fig.4 respectively show the evolution of the filters’
RMS resulting from the assimilation run for the
physical and the ecological variables, and compare
them to those obtained from the model free-run.
For both sub-systems, the dual assimilation clearly
enhances the models fit to the data for all variables
and throughout the assimilation window. After a
large reduction of the estimation error at the first
analysis step, subsequent filters’ analyses are less
important.
able to improve the overall behavior of both subsystems. However, more experiments are needed
to closely assess the impact of omitting the impact
of assimilating the ecological data into the
physical model, which can be important when real
data are assimilated to guaranty a consistency
between both filters analyses.
Fig. 4. RMS for ecological variables.
Fig. 3. RMS for physical variables.
Overall, the state variables estimates were
improved by almost 70% for the physical model
and by more than 45% for the ecological model.
Better results could have been obtained for the
ecology if the SEEK filter was used, but this would
have entailed significant increase in the total
computational cost. The assimilation results also
suggest that the analyses of the two sub-systems
are consistent, as no unstable behavior was
detected in the evolution of both filters forecasts. In
other experiments not shown here, the assimilation
of SSH alone was also found to have an important
impact on the ecology, although additional
assimilation of CHL data leads to further
improvements. This is in accordance with the
oligotrophic conditions that characterize the system
where the evolution of the ecological variables
strongly depend on the physical forcing, and shows
the importance of improving the physical forcing for
the estimation of the ecology.
Comparing to the joint approach, the dual
approach respects more the one-way coupling of
our ecosystem model. It also offers more flexibility,
e.g. more degrees of freedom to fit the data,
different choices for the parameters of the
assimilation scheme according to the need of each
sub-system, which is very important for an efficient
implementation, in term of cost and performance.
In our experiments, the dual filtering system was
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