The methods of data assimilation for sea dynamics problems are very important and are studied by many authors. We present our research on the development of variational data assimilation technique for sea dynamics problems with the aim to correct or to find the sea surface heat fluxes using the observational data. We consider the mathematical model of sea dynamics developed at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS). For observational data, we use daily mean SST observations from the Copernicus Marine Data Store. The variational data assimilation technique is based on a minimization of the cost function related to the observational data on the sea surface. The cost function involves the background and observation error covariance matrices. Our novelty is that the problem is reduced to a coupled system of model and adjoint equations, and it is solved by an iterative algorithm with the optimal iterative parameters. Numerical experiments for the Baltic Sea circulation model demonstrate the efficiency of the developed methodology. It is shown that the use of the developed data assimilation technique makes it possible to bring model calculations closer to observational data, and thereby helps to improve the forecast properties of the model.

In recent years, methods for studying and numerically solving problems of variational data assimilation, which are specific problems of optimal control, have been greatly developed in meteorology and oceanography, where observational data are assimilated in atmospheric and ocean models to obtain initial boundary conditions or other model parameters for subsequent modeling and forecasting. This paper considers the variational data assimilation algorithm for the sea dynamics model developed at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences. This problem is formulated as an optimal control problem. An optimality system includes a direct equation, adjoint equations, satellite observations, and covariance matrices of observation and background errors. As an application, a mathematical model of the Black Sea dynamics with a block of variational assimilation of data on the sea surface temperature is considered. The data source in the proposed study is the Aqua satellite with the MODIS spectrometer and the SNPP satellite with the VIIRS spectrometer. A feature of these data is that they do not cover the entire study area, and the assimilation procedure uses a characteristic function that determines data availability at the time of assimilation. It is shown that introducing the data assimilation procedure into the model makes it possible to obtain sea surface temperature calculation results closer to the observed ones, improving the predictive properties of the numerical model.​

The use of Four-Dimensional variational (4D-Var) data assimilation technology in the context of sea dynamics problems, with a sensitivity analysis of model results to observation errors, is presented. The technology is applied to a numerical model of ocean circulation developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), with the use of the splitting method and complemented by 4D-Var data assimilation with covariance matrices of background and observation errors. The variational data assimilation involves iterative procedures to solve inverse problems so as to correct sea surface heat fluxes for the model under consideration. An algorithm is formulated to study the sensitivity of the model outputs, considered as output functions after assimilation, to the observation errors. The algorithm reveals the regions where the output function gradient is the largest for the average sea surface temperature (SST) in a selected area, obtained by assimilation. In the numerical experiments, a 4D variational problem of SST assimilation for the Baltic Sea area is solved.

The technology aimed at high-performance computing is presented for modeling the sea dynamics problems based on 4D variational data assimilation technique developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS). The technology is based on the multicomponent splitting method for the mathematical model of sea dynamics and the minimization of cost functionals related to the observation data by solving an optimality system that involves the adjoint equations with observation data and observation error covariances. Efficient algorithms for solving the variational data assimilation problems are presented based on modern iterative processes with a special choice of iterative parameters. The technology is illustrated for the Baltic Sea dynamics model with variational data assimilation to restore the initial states and the heat fluxes on the sea surface.

The 4D variational data assimilation technique is presented for modelling the sea dynamics problems, developed at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS). The approach is based on the splitting method for the mathematical model of sea dynamics and the minimization of cost functionals related to the observation data by solving an optimality system that involves the adjoint equations and observation and background error covariances. Efficient algorithms for solving the variational data assimilation problems are presented based on iterative processes with a special choice of iterative parameters. The technique is illustrated for the Black Sea dynamics model with variational data assimilation to restore the sea surface heat fluxes.

2 Московский государственный университет имени М. В

This work is aimed at using the marine data of the Shared Use Centre (SUC) “IKI-Monitoring” in the variational assimilation procedures of the Informational Computational System (ICS) “INM RAS - Black Sea”. SUC “IKI - Monitoring” is a tool for obtaining remote sensing observations on the Earth state. In the paper observation data information is given, data processing procedures are described, algorithms for the assimilation of the information received and several specific features of the numerical model used are presented. Results of the variational assimilation of two sets of observation data are presented and discussed. Numerical experiments have confirmed the possibility of using incomplete data from satellites in the problems of modelling the sea area.

The problems of modeling hydrothermodynamics of particular sea and coastal areas are of current interest, since the results of this modeling are often used in many applications. One of the methods allowing to take into account open boundaries and bring the simulation results closer to real data is the variational assimilation of observational data. In this paper the following approach is considered: it is supposed that there are observational data at a certain moment in time; the problem is considered as an inverse problem, in which the functions of fluxes across the open boundary are treated as additional unknowns. Comparison of methods for reconstructing unknown functions in boundary conditions at an open boundary using sea level and velocity observational data in a number of numerical experiments for a region of a simple shape is carried out.

The formulation of boundary conditions at liquid (open) boundaries is a topical problem in mathematically modeling the hydrothermodynamics of open water areas. Variational data assimilation is one method allowing one to take into account liquid boundaries in models. According to the approach considered in this paper, observational data at a certain time are given and the problem is treated as an inverse one with open boundary flows as additional unknowns. This paper presents a formulation of the general problem of the variational assimilation of observational data for a model of the hydrothermodynamics of open water areas based on the splitting method. Algorithms for the variational assimilation of temperature and sea-level data at the liquid boundary are formulated and the results of numerical experiments on the use of the algorithms in the Baltic Sea circulation model are presented.

A series of problems related to the class of inverse problems of ocean hydrothermodynamics and problems of variational data assimilation are formulated in the present paper. We propose methods for solving the problems studied here and present results of numerical experiments.

A new technique is proposed for constructing domain decomposition algorithms based on optimal control theory, the theory of inverse and ill-posed problems, application of adjoint equations, and modern iterative processes. The technique applies to a broad class of problems in mathematical physics (including ones with nonsymmetric operators, convection-dominant operators, etc.).

The technology is presented for modeling and prediction of marine hydrophysical fields based on the 4D variational data assimilation technique developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS). The technology is based on solving equations of marine hydrodynamics using multicomponent splitting, thereby solving an optimality system that includes adjoint equations and covariance matrices of observation errors. The hydrodynamic model is described by primitive equations in the sigma-coordinate system, which is solved by finite-difference methods. The technology includes original algorithms for solving the problems of variational data assimilation using modern iterative processes with a special choice of iterative parameters. The methods and technology are illustrated by the example of solving the problem of circulation of the Baltic Sea with 4D variational data assimilation of sea surface temperature information.

Purpose. In order to simulate the sea hydrothermodynamics, the problem of variational assimilation of the sea surface temperature data is solved. The data assimilation permits to adjust the numerical model calculations to the measurement data obtained in the environment under study. Methods and Results. The mathematical model of hydrothermodynamics of the Black and Azov seas, developed at the Institute of Numerical Mathematics of RAS and written in the sigma coordinate system, is considered. The distinctive feature of the model consists in applying the splitting method to physical processes and spatial coordinates that can significantly simplify the variational data assimilation algorithm. The problem of variational assimilation of the sea surface temperature data is formulated. A cost functional has been introduced; it includes the control function - heat flux at the sea upper boundary and satellite observations of the sea surface temperature. The necessary condition for the functional minimum is reformulated through the optimality system including the direct and adjoint problems, and the control condition. Using the variational assimilation of the satellite-derived observations, the algorithm for solving the stated problem was developed. It takes into account the observational errors' covariance matrix calculated based on the statistical characteristics of the sea surface temperature observational data. The algorithm implies a sequential solution of the optimality system in the iterative process with the specially selected iterative parameter. The results of numerical solution of this problem are represented by the example of the Black and Azov seas. Conslusion. The results of numerical modeling with the observational data assimilation and without it are compared; efficiency of the observational data assimilation procedures is shown. Influence of the sea surface temperature assimilation upon the other system parameters is investigated. It is shown that when assimilating the sea surface temperature, only temperature in the upper layers is affected, whereas, provided that the depth is sufficient, the profile in the lower layers remains practically unchanged. The impact on the other system parameters is either minimal or not manifested at all.

Recently, significant results have been achieved in studying and modeling the processes of large-scale sea and ocean variability.This is due to changes in the level of development of tools, methods and equipment.These include: new observational systems (satellites, ARGO buoys, etc.), information analysis methods and numerical algorithms, powerful computers that allow processing large information flows of calculations and observations.We consider the problem of four-dimensional variational data assimilation of satellite data on the sea surface temperature using a model of sea hydrothermodynamics developed at INM RAS in this paper.Problem state and methods for its solution are discussed, the results of numerical experiments with real data of satellite observations are presented, including cases when the observation data are known at the part of the water area.

One of the modern fields in mathematical modelling of water areas is developing hybrid coastal ocean models based on domain decomposition. In coastal ocean modelling a problem to be solved is setting open boundary conditions. One of the methods dealing with open boundaries is variational data assimilation. The purpose of this work is to apply the domain decomposition method to the variational data assimilation problem. The method to solve the problem of restoring boundary functions at the liquid boundaries for a system of linearized shallow water equations is studied. The problem of determining additional unknowns is considered as an inverse problem and solved using well-known approaches. The methodology based on the theory of optimal control and adjoint equations is used. In the paper the theoretical study of the problem is carried out, unique and dense solvability of the problem is proved, an iterative algorithm is proposed and its convergence is studied. The results of the numerical experiments are presented and discussed.

В настоящей работе приведены результаты численного решения задачи вариационной ассимиляции данных об уровне на жидкой (открытой) части границы в модели гидротермодинамики Балтийского моря.Под жидкими границами акватории подразумеваются границы раздела морей и океанов, а также проходящие по проливам, устьям рек и т. д.Задание граничных условий на жидких границах является важной проблемой современной геофизики.Одним из существующих методов, которые можно применить для учёта жидких границ в моделях, является использование вариационной ассимиляции данных наблюдений, в том числе информации об уровне.Так, имея данные наблюдений в некоторый момент времени, можно поставить обратную задачу о восстановлении потоков через открытую границу.В работе приведена постановка задачи вариационной ассимиляции данных об уровне на жидкой границе, сформулирован итерационный алгоритм её решения, а также некоторые выводы о сходимости алгоритма и разрешимости исходной задачи.Подробно рассмотрены результаты применения алгоритма к решению задачи моделирования гидротермодинамики Балтийского моря, а так

The results of the development of the Informational Computational System (ICS) "INM RAS - Baltic Sea" are presented. The System includes numerical model of the Baltic Sea hydrothermodynamics, the oil spill model describing the propagation of a slick at the sea surface with an assessment of the risk of pollution of coastal areas and the optimal ship route model. The ICS is based on the INMOM numerical model of the Baltic Sea hydrothermodynamics and includes data assimilation procedures. It is possible to calculate the main hydrodynamic parameters (temperature, salinity, velocities, sea level) using user-friendly interface of the ICS. Main possibilities of the ICS are presented in the work.