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A sequential data assimilation method based on the properties of a
diffusion-type process and its applications with the ocean models
MOM4 and HYCOM
Clemente A. S.
1,2
Tanajura
and Konstantin
1,3
Belyaev
1Centro
de Pesquisa em Geofísica e Geologia (CPGG/UFBA)
2Dept. de Física da Terra e do Meio Ambiente, Instituto de Física (IF/UFBA)
Universidade Federal da Bahia, Salvador, Brazil (UFBA)
3Shirshov Institute of Oceanology, Russian Academy of Sciences (Shirshov)
1. Introduction
Two different applications of the data
assimilation scheme were done. In the first, the
Hybrid Ocean Coordinate Model (HYCOM) was
used to assimilate PIRATA data, and in the second
the Geophysical Fluid Dynamics Laboratory
(GFDL/NOAA) Modular Ocean Model v4
(MOM4) MOM4 was used to assimilate
TAO/TRITON data. Only vertical profiles of
temperature were used and just a few assimilation
steps were realized in each experiment. The goals
of these experiments are to verify the feasibility of
the scheme and to face the model biases and
difficulties in the numerical realization process.
The present data assimilation scheme may be
used as part of a numerical ocean forecast system
for the Atlantic Ocean, with emphasis over the
region along the Brazilian Coast that is today under
construction by the Oceanographic Modeling and
Observation Research Network (REMO). This
effort is supported by Petrobras and the Brazilian
Agency for Petroleum, Natural Gas and Biofuels.
2.1 HYCOM
HYCOM was configured for the Atlantic Basin
using a 1/3o spatial grid and 22 vertical hybrid
layers with the top layer at its minimum thickness
of 3m. In coastal waters, there are up to 12 sigma
levels and the coastline is at 10m isobath. North
and south boundaries for temperature, salinity and
pressure are linearly relaxed toward the Levitus
seasonal climatological data (Levitus and Boyer,
1994).
Initially, the model was run for a 30-year
period forced with COADS climatological
atmospheric fields in order to produce a
climatological ocean state. The fields used to force
HYCOM were the surface air temperature, surface
air specific humidity, precipitation rate, net
shortwave radiation, net radiation and wind stress.
Fig. 1 compares the temperature annual mean at
the equator for the last year of the climatological
run versus Levitus data. It shows that this HYCOM
run was not able to produce a well-defined
thermocline region despite the temperature at sea
surface and well below the thermocline being
close to climatology.
After the 30-year run, the 6-hour 2007 NCEP
reanalysis atmospheric forcing fields were used to
produce more realistic oceanic conditions for
actual time. The model was run sequentially for 4
years with the 2007 atmospheric reanalysis data to
force the model towards the 2007 ocean state.
For the realization of the assimilation scheme,
the covariance matrix of observational data γij was
prescribed as a function of the area averaged
temperature and the distance between each pair of
observations. Constant C presented in the poster by
Belyaev and Tanajura was estimated as the
difference between the spatial average among the
available observations and the model at each grid
point .
UFBA
Fig. 3 shows a detail of the top 300 m at the
equator for the same day (Jan 10, 2008) for the
simulation and assimilation runs. The assimilation
causes a deepening of the mixed layer and the
increase of the vertical gradient of temperature in the
thermocline region. An unstable profile is produced
around 50 m depth between 25oW and 30oW, showing
that realization of data assimilation in HYCOM still
needs further developments. For instance, a rigorous
quality control of the input data is necessary.
Data assimilation methods are important tools
for numerical weather and climate forecasts, for
diagnostic studies and to complement monitoring
offered by observational systems in the oceans and
the atmosphere (e.g. Kalnay 2003). In the present
study, a new formulation for sequential data
assimilation schemes is proposed based on the
mathematical properties of a diffusion-type process.
The detailed mathematical formulation of the
method and the conditions for the convergence of
the finite dimensional distributions of random
process into the distribution of a stochastic process
are given in Tanajura and Belyaev (2008) and in
the poster presented by Belyaev and Tanajura in the
present workshop. The method may be reduced to
the well-known optimal interpolation scheme the
model is unbiased and all covariances are known a
priori. Also, this scheme may be reduced to the
linear Kalman-Bucy filter when observational data
have Gaussian noise.
2. Numerical Experiments
UFBA
Fig. 1. Vertical structure of the annual mean of
temperature at the equator for the last year of the 30year climatological run and Levitus climatology. (oC)
All assimilation was performed in each model
vertical layer independently. The PIRATA daily data
used in the assimilation were converted into
potential temperature and vertically interpolated to
each model hybrid layer depth. Before this
procedure, data was passed quality control tests to
avoid assimilation of spurious data.
Despite using temperature data only, salinity
was directly affected by the assimilation procedure.
After having the temperature ammendment, salinity
was re-calculated through the state equation
maintaining the model density unchanged. To
perform the data assimilation scheme more
rigorously the analysis increment was added to the
model background in smaller parts. In the present
realization, two steps were taken. First, the model
error with respect to observations was calculated.
Then, this error was divided by two and the
assimilation increment was calculated twice in a
row. This procedure can be realized for any number
of intervals until a maximum restricted by the model
time step.
The results presented here deal with the daily
assimilation of PIRATA data from January 1 until
January 10, 2008. HYCOM was initialized with the
last output produced by the 2007 run, and it was
forced with 6hr NCEP reanalysis data.
Fig. 2 shows the potential temperature at 30 m
produced by the model simulation (without
assimilation), the assimilation run, and the
difference assimilation minus simulation on January
10, 2008. The assimation run imposes corrrections in
the PIRATA region, with a warming of about 2oC
between 20oN and 30oN and cooling between the
equator and 20oS.
Fig. 3. Vertical structure of temperature at the equator for
Jan 4, 2008 from simulation and assimilation runs.
The data assimilation reduced the difference
between model and observation along the 10-day
integration (Fig. 4).
Fig. 6. MOM4 simulation on Jan 2, 2001 of 5 m depth
temperature in contour lines and mooring locations
marked by shaded circles (top); assimilation;
difference assimilation minus simulation (bottom)(oC).
Fig. 4. Temperature
RMSE time
evolution (oC) from
Jan 1 until Jan 10
2008 before (solid
line) and after
assimilation (dashed
line).
2.2 MOM4
For the MOM4 experiment, the model was also
run for 30 years from rest and Levitus temperature
and salinity, and then forced from Jan 1995 until Dec
2000 with monthly mean data from the Common
Ocean-ice
Reference
Experiments
(CORE)
(http://data1.gfdl.noaa.gov/nomads/forms/mom4/CO
RE.html). The model was configured with a global
grid of 1o resolution, except in between 10oS – 10oN
in which the resolution was 1/3o, and between 10o –
30o in both hemispheres where the resolution was
linearly increasing from 1/3 to 1o .
Fig. 7. Temperature vertical profiles by the model
simulation (thin line), assimilation (thick line) and
observation (dashed line) at the points (0°N, 180°E) and
(2°N, 190°E) on January 2, 2001. Unit is oC.
3. Conclusions
The results presented here show the data
assimilation scheme was successfully applied in
different models and it works properly. However,
the correct procedure to assimilate vertical profiles
of temperature in HYCOM is still not completely
dominated, mainly because of the difficulties in
interpolating the in situ temperature in the
corresponding model isopicnal layers.
Acknowledgements.
This work was financially
supported by PETROBRAS and Agência Nacional do
Petróleo, Gás Natural e Biocombustíveis (ANP), Brazil,
via the Oceanographic Modelling and Observation
Research Network (REMO).
Fig. 5. Vertical structure of temperature at the equator for
the last year of MOM4 climatological run. (oC)
Fig. 5 shows the annual mean of the potential
temperature at the equator. Comparing with Fig. 1, the
simulation of MOM4 is able to resolve more
accurately the vertical structure of temperature than
HYCOM.
Fig. 6 shows the model simulation, assimilation
and the difference for 5 m depth temperature on
January 2, 2001. The model is mostly corrected with
positive increments of temperature at the surface.
Fig. 7 shows the vertical profiles of temperature at
two points, one in which there is a mooring and the
Fig 2. January 10, 2008 temperature for (a) the HYCOM other in which the average of
neighbooring
simulation; (b) assimilation; and (c) the difference moorings.
o
assimilation minus simulation. Unit is C.
References
Kalnay, E. 2003. Atmospheric Modeling ,
Predictability and Data Assimilation. Springer.
Tanajura, C.A.S. and K. Belyaev. A sequential data
assimilation method based on the properties of a
diffusion-type process. Applied Mathematical Modelling
(in press) (2008)