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A sequential Data Assimilation Method Based on
Diffusion Approximation General and Specific
Aspects Konstantin Belyaev1,2 and Clemente A. S.
Tanajura1,3 1Centro de Pesquisa em Geofísica e
Geologia (CPGG/UFBA) 2Shirshov Institute of
Oceanology, Russian Academy of Sciences
(Shirshov) 3Dept. de Física da Terra e do Meio
Ambiente, Instituto de Física (IF/UFBA) Universida
de Federal da Bahia, Salvador, Brazil (UFBA)
.
.
1. Introduction The correction of the model
results by observational data by a data
assimilation scheme is generally realized as
follows. Given a time-interval (to ,T), it can be
broken down into subintervals (to , t1), (t 1,
t2),..., (tk,tk1). On any subinterval (tk,tk1),
the model starts at moment tk with the initial
vector ?k , also called as background state
vector, and integrates forward until moment tk1
, when it produces a new state-vector
.Here and further, the superscript m
indicates that the system state has been obtained
only by model integration without any other
source of information. Over the time-interval
(tk,tk1), a series of measurements of the state
variable represented by the vectors are taken
independently of the model. Then, the model
output at time k1 is corrected by both
observational information and the model state
taken during the time interval according to the
scheme
() Here, H denotes the
projection of model variables in the model domain
onto the observational locations at each
subinterval. The model and the measured variables
are considered at the moments in which the
observational data become available. The weight
functions a also depend on time and space and may
be either known a priori or determined by some
appropriate algorithm. The corrected
state-vector, the so-called objective analysis,
is now taken as the new initial condition for the
next step of the model integration. In spite of
substantial differences in the calculation of the
weight coefficient, the majority of the data
assimilation methods follows the scheme mentioned
above. Therefore, here we investigate several
general aspects and mathematical properties of
this scheme. Also, some numerical applications
were performed and few results are presented.
3. The Optimisation Problem Based on the theorem
above the following optimization problem can be
posed and solved Find the weight coefficients a
so that they minimise the functional under the
condition where C is a constant associated with
the model bias with respect to observations. The
minimisation of this functional leads to a as a
solution of the equations where F is an
unknown vector of Lagrangean multipliers. 4.
Results To show an application of the method in
oceanography, the model MOM4 was used along with
observational data of vertical profiles of
temperature from the TOGA/TAO array. Few
results are presented below for the assimilation
of daily data on January 2, 2001 in which the
model was forced with NCEP reanalysis. All
details are described in Tanajura and Belyaev
(2008).
Fig. 1 shows the model simulation, assimilation
and the difference for 5 m depth temperature on
January 2, 2001. Fig. 2 shows the vertical
profiles of temperature at two points, one in
which there is a mooring and the other in which
the average of neighboor moorings lying within
a circle with 1000 km radius was taken as an
estimate of the observations.
Knowing (i) the solution of system (1) for the
given initial vector on the entire
interval (ii) the observed variables
and (iii) the value of the random index ?k,n
, the newly constrained variables are introduced
by the formula
(2) In
this manner, it becomes possible to constrain the
sequence of trajectories of the random process
over the entire interval (0,T). Starting from
some known random vector , the solution of
(2) on each interval ?tk,n with jumps at the
correction time can be evaluated. The limiting
behaviour of the trajectories provided by (2)
when is here investigated. Theorem Le
t the following conditions hold A1. The
intervals ? 0 uniformly with respect to k,
i.e.,
A2. The probability
distribution for the random variables ?k,n
satisfies the conditions for any k, and
A3. The random
vectors have 2? moments for positive ? in each
series n, i.e.,
, and these
variables are uniformly bounded with respect to n
for any k A4. The operator ? is a continuous
function on its arguments A5. The set of weight
coefficients is uniformly bounded with respect to
n A6. There exists some random variable and
the sequence of distributions of random variables
converges to the distribution of , i.e., for
each x as n??. Then, the sequence of
finite-dimensional distributions of random
processes converges to
the distribution of a stochastic process ,
which will be a solution of the following
stochastic differential equation where and
The standard Wiener process w(t) is defined on
the interval (0,T) and it is independent of the
random variable .
a)
b)
Figure 2. Temperature vertical profiles
calculated by the model simulation (thin line),
assimilation (thick line) and observation (dashed
line) at the point (a) (0N, 180E) and (b)
(2N, 190E) on January 2, 2001. Unit is oC.
5. Conclusions The results presented here are of
both theoretical and practical interests. From
the theoretical point of view, it is important to
know (i) when and under which conditions the
solution of differential equations (1) with the
correction (2) may be approximated by a
continuous function and (ii) how the limit
function can be calculated numerically. From the
practical point of view, the scheme can be
utilised to determine a variety of relevant
parameters, such as a measure of the assimilation
quality. Also, the proposed scheme is rather
general and, therefore, some popular schemes,
such as optimal interpolation, can be considered
as special cases. 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).
a)
b)
References Daley, R. Atmospheric Data Anaslysis.
Cambridge Univ. Press. 457 pp. (1991) Kalnay, E.
Atmospheric Modeling, Data Assimilation and
Predictability. Cambridge Univ. Press, 341 pp.
(2003) Tanajura, C.A.S. and K. Belyaev. On the
oceanic impact of a data assimilation method on a
coupled ocean-land-atmosphere model. Ocean
Dynamics, 52, 123-132 (2002). 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)
c)
Figure 1. (a) MOM4 model simulated 5 m depth
temperature field in contour lines and mooring
locations marked by shaded circles (b)
assimilated 5 m depth temperature field (c)
difference assimilation minus simulation at 5 m
depth temperature. The simulation is for January
2, 2001. Unit is oC.