Physical Oceanography, Vol. 14, No. 6, 2004
MATHEMATICAL MODELING OF MARINE SYSTEMS
DYNAMIC-STOCHASTIC MODELING OF NATURAL PROCESSES
E. M. Igumnova and I. E. Timchenko
We propose a method for the construction of dynamic-stochastic models of natural systems
based on the assimilation of the data of observations in the prognostic equations of coupled pro-
cesses. In these models, the method of adaptive balance of causes is used to deduce evolutionary
equations of the analyzed processes and assimilate the data of observations in these equations.
The deduced general equations are considered for an example of a marine ecosystem character-
ized by the development of four coupled processes. It is shown that the optimal prediction of
these processes requires the solution of 11 systems of equations with simultaneous adaptation of
prognostic estimates and the coefficients of the models to the data of observations. A numerical
simulation experiment explaining the algorithm of the proposed method of modeling is consi-
dered. A conclusion is made that the application of this method in the geoinformation systems of
monitoring of the environment is quite promising.
Dynamic-stochastic models (DSM) of the ocean are based on the equations of dynamics of the ocean sup-
plemented with stochastic algorithms of assimilation of the data of observations. Most often, these models are
used for the construction of diagnostic and prognostic maps of the fields of parameters of the ocean. The basic
principle of construction of DSM is reduced to averaging of the equations of dynamics of the ocean (conditional
with respect to the data of observations) [1, 2]. As an algorithm of interpolation of the data of observations for
the construction of DSM in practice, it is customary to use the Kolmogorov theory of interpolation of stationary
random functions . Note that this theory serves as a basis for the development of the methods aimed at the
objective analysis of the fields of parameters of the ocean .
In Kolmogorov’s theory, it is supposed that the measurements of a random process generate conditional
probability distributions in the sections of its ensemble of realizations. The mean values of these distributions
are the best approximations to the actual processes (in the class of linear estimates). To find these values, it is
necessary to interpolate the data of observations with the optimal weight coefficients, which can be determined
from the Kolmogorov system of equations if the correlation function of the analyzed process is known. The best
estimate of the value of a field at a given point is delivered by the sum of two terms weighted with certain
weights: the prediction of the field according to the dynamic model and the “discrepancy” of this prediction, i.e.,
the result of interpolation (at this point) of the field of errors of the prediction relative to the data of observations.
The outlined approach to the construction of the optimal estimate of a random process can be generalized to
systems of coupled natural processes. In the objective analysis of the fields of parameters of the ocean, a similar
problem of “statistical adjustment of fields” was solved in a series of works [4, 5]. At present, the problem of
statistical adjustment of coupled processes becomes especially significant in connection with the problem of
modeling of complex ecological-economic systems, such as, e.g., “sea–dry land” natural-economic complexes
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol. Translated from Morskoi Gidrofizicheskii Zhurnal,
31–42, November–December, 2004. Original article submitted April 2, 2003.
348 0928-5105/04/1406–0348 © 2005 Springer Science+Business Media, Inc.