Correction of Systematic Error and Estimation of Confidence Limits for one Data Assimilation Method
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Correction of Systematic Error and Estimation of Confidence Limits for one Data Assimilation Method K. P. Belyaev1, 2* , A. A. Kuleshov3** , and N. P. Tuchkova2*** (Submitted by A. V. Lapin) 1
Shirshov Institute of Oceanology of Russian Academy of Sciences, Moscow, 117218 Russia Federal Research Center “Computer Sciences and Control,” Russian Academy of Sciences, Moscow, 119333 Russia 3 Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, 125047 Russia 2
Received March 19, 2020; revised April 2, 2020; accepted April 10, 2020
Abstract—The paper studies the properties of the previously proposed method for assimilating observational data into a hydrodynamic model, which is an author’s version of the generalized Kalman filter. This method generalizes the well-known ensemble Kalman filter method. The equations of the generalized Kalman filter are extended to the case when the initial hydrodynamic model is biased relative to the observations, that is, it has a systematic error. In addition, the problem of estimating the confidence limits of the model variables (analysis) constructed after assimilation is considered. The corresponding Fokker–Planck–Kolmogorov equation for these estimates is given. For a special case, which is usually encountered in practice, an analytical solution of this equation is given by the perturbation theory method. Numerical examples are also carried out for a specific dynamic model, and an analysis is discussed of these calculations. DOI: 10.1134/S1995080220100054 Keywords and phrases: dynamical system, data assimilation theory, Generalized Kalman filter, Fokker–Planck–Kolmogorov equation.
1. INTRODUCTION The work continues the study of the previously proposed data assimilation (DA) method, called by the authors as the one of the method of the Generalized Kalman filter (GKF) [1, 2]. It should be said that the task of observational DA into a dynamic system (model) defined by a system of differential and/or integro-differential equations is one of the most urgent problems of modern oceanology, geophysics, meteorology and climatology and uses methods such as classical mathematics, for instance, such mathematical disciplines as the theory of optimal control, mathematical statistics, probability theory and mathematical physics, as well as methods of the latest developments, such as information systems, parallel programming, large databases, and a number of others. Moreover, in contrast to data interpolation, the DA problem(s) involves changing all the characteristics of a dynamic system, both observable and unobservable, in order to maintain a balance between the model parameters, which usually consists in mathematical notation of conservation laws. When assimilating the data, it is required to maintain the balance conditions and at the same time bring the model solution closer to the observations in the given metric. The studies in this area are ongoing since the end of 60s of the last century and well known from the scientific literature. A good review of existing theoretical and
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