Journal of Mathematical Sciences, Vol. 111, No. 3, 2002 A DATA ASSIMILATION METHOD IN A NUMERICAL MODEL AND ITS APPROXIMATION BY A STOCHASTIC DIFFUSION PROCESS K. Belyaev and C. A. S. Tanajura (Petr´ opolis, Brazil) UDC 519.2 1. Introduction In recent years, advances have been made in computational facilities and observational data systems. The devel- opment of ocean general circulation models (OGCMs) and atmospheric general circulation models (AGCMs), as well as of coupled ocean–land–atmosphere general circulation models (CGCMs), have led to improvements in the analysis of the environment and climate prediction. Data revolution in oceanography is bringing the daily practice of physi- cal oceanography closer to dynamic meteorology. Reliable data assimilation techniques are needed for OGCMs and CGCMs to fully exploit the new observational facilities. Data assimilation provides a better estimate of the ocean and atmospheric physical states. It has been developed to extract as much information as possible from both measure- ments and dynamic models by combining them in an optimal way. Assimilation may be used to improve initial and/or boundary conditions, and to evaluate poorly known model parameters. Two general concepts on the mathematical formulation of data assimilation methods have been discussed in the literature. The first is the variational/adjoint method, which has been a popular scheme (see, e.g., [1]). As an example of this technique, let the model initial and/or boundary condition be unknown and the observational data be distributed over some time interval. The technique seeks an optimal initial and/or boundary condition with respect to some criteria by comparing the model trajectory with measurements. This can be formulated as a constrained minimization problem. The present paper does not deal with this class of methods. The other class of methods is the sequential data assimilation. Starting from some initial condition, the model solution is sequentially updated whenever measurements are available. The model solution approaches the observed state under certain conditions. This class of methods requires an updating scheme, which combines the model solution and the measurements to find the “best” state estimate. The Kalman-filter approach represents this class of methods. The method, presented in [2], is based on Kalman-filter theory. It uses the phase-space representation and the Fokker–Planck equation for the probability density. Its usefulness and technical advantages have been demonstrated. Nevertheless, some important theoretical questions remained without consideration. In particular, the necessary and sufficient conditions to apply the method and approximate the numerical solution by a stochastic diffusion process were not discussed. Also, the stability of the scheme was not fully studied. In the present note, some of these issues are considered. It is proved that the time-sequence of model solutions corrected according to [2] converges to the stochastic diffusion process with known drift and diffusion parameters. Also, the stability of the method is shown when these parameters are defined with perturbations. The method of the proof of convergence is based on the general theory of convergence of Markov chains to stochastic diffusion processes (e.g., [3]), but exploits specific properties of the model. Also, the method of the proof of stability uses Fourier transformation and corresponding estimations for distributions from the Berry–Esseen inequality. This paper contains results of some numerical experiments with an OGCM. The goals are to demonstrate the practical usefulness of the data assimilation method and its approximation by a diffusion stochastic process. This study uses data from the Pilot Research Moored Array in the Tropical Atlantic (PIRATA) observational project, which is being carried out in the tropical Atlantic (see, e.g., [4]) by France, the USA, and Brazil. The observational array is formed by 12 moored buoys, which record ocean surface and subsurface temperatures and other surface meteorological quantities. PIRATA gives a unique opportunity to improve and validate dynamic models and data assimilation techniques by comparing numerical results with in situ measurements. In the numerical experiments performed here, the Fokker–Planck equation is solved to define the probability distribution of the sought variable (namely, ocean temperature) error. The observed variable should lie within the boundaries determined by the solution of the Fokker–Planck equations. Experiments confirm this property. Proceedings of the Seminar on Stability Problems for Stochastic Models, Eger, Hungary, 2001, Part I. 1072-3374/02/1113-3505$27.00 c  2002 Plenum Publishing Corporation 3505