Methods for assimilation of observational data in problems of the physics of the atmosphere and the ocean

In this paper we review and analyze approaches to data assimilation in geophysical hydrodynamics problems, starting with the simplest successive schemes of assimilation and ending with modern variational methods. Special attention is paid to the the study of the problem of variational assimilation in the weak formulation and construction of covariance error matrices of the optimal solution. This is a new direction, to which the author made a contribution: an optimality system is formulated for the problem of variational data assimilation in a weak formulation and algorithms for deriving the covariance error matrices of the optimal solution are developed.

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Sensitivity of functionals of variational data assimilation problems

Доклады Академии наук ◽

10.31857/s0869-56524864421-425 ◽

2019 ◽

Vol 486 (4) ◽

pp. 421-425

Author(s):

V. P. Shutyaev ◽

F.-X. Le Dimet

Keyword(s):

Data Assimilation ◽

Observational Data ◽

Response Function ◽

Optimal Solution ◽

Evolutionary Model ◽

Variational Data Assimilation ◽

Adjoint Equations ◽

Unknown Parameters ◽

Initial State ◽

Nonstandard Problem

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to find simultaneously unknown parameters and the initial state of the model. The response function is considered as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. On the basis of the Hessian of the original cost function, the solvability of the nonstandard problem is studied. An algorithm for calculating the gradient of the response function with respect to observational data is formulated and justified.

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Studying the sensitivity of the optimal solution of the variational data assimilation problem for the Baltic Sea thermodynamics model

Russian Meteorology and Hydrology ◽

10.3103/s1068373915060072 ◽

2015 ◽

Vol 40 (6) ◽

pp. 411-419 ◽

Cited By ~ 1

Author(s):

V. P. Shutyaev ◽

E. I. Parmuzin

Keyword(s):

Data Assimilation ◽

Baltic Sea ◽

Optimal Solution ◽

Variational Data Assimilation ◽

The Baltic Sea ◽

The Baltic

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Adjoint equations in variational data assimilation problems

Russian Journal of Numerical Analysis and Mathematical Modelling ◽

10.1515/rnam-2018-0012 ◽

2018 ◽

Vol 33 (2) ◽

pp. 137-147 ◽

Cited By ~ 2

Author(s):

Victor P. Shutyaev

Keyword(s):

Data Assimilation ◽

Iterative Algorithms ◽

Variational Data Assimilation ◽

Evolution Model ◽

Adjoint Equations ◽

Optimality System ◽

Initial Condition

Abstract The problem of variational data assimilation for an evolution model is considered with the aim to identify the initial condition. The solvability of the optimality system is studied. Based on the adjoint equations, iterative algorithms for solving the problem are developed and justified.

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Sensitivity analysis with respect to observations in variational data assimilation for parameter estimation

Nonlinear Processes in Geophysics ◽

10.5194/npg-25-429-2018 ◽

2018 ◽

Vol 25 (2) ◽

pp. 429-439 ◽

Cited By ~ 2

Author(s):

Victor Shutyaev ◽

Francois-Xavier Le Dimet ◽

Eugene Parmuzin

Keyword(s):

Data Assimilation ◽

Response Function ◽

Optimal Solution ◽

Coupled System ◽

Variational Data Assimilation ◽

Adjoint Equations ◽

Observation Data ◽

Unknown Parameters ◽

The Baltic ◽

Nonstandard Problem

Abstract. The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a nonstandard problem involving the coupled system of direct and adjoint equations. The nonstandard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is presented. A numerical example is given for the variational data assimilation problem related to sea surface temperature for the Baltic Sea thermodynamics model.

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Sensitivity with respect to observations in variational data assimilation

Russian Journal of Numerical Analysis and Mathematical Modelling ◽

10.1515/rnam-2017-0006 ◽

2017 ◽

Vol 32 (1) ◽

Cited By ~ 3

Author(s):

Victor Shutyaev ◽

Francois-Xavier Le Dimet ◽

Elena Shubina

Keyword(s):

Data Assimilation ◽

Response Function ◽

Nonlinear Evolution ◽

Function Analysis ◽

Optimal Solution ◽

Coupled System ◽

Variational Data Assimilation ◽

Adjoint Equations ◽

Observation Data ◽

Standard Problem

AbstractThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. The solvability of the non-standard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is developed and justified.

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Accounting for Model Error in Variational Data Assimilation: A Deterministic Formulation

Monthly Weather Review ◽

10.1175/2010mwr3192.1 ◽

2010 ◽

Vol 138 (9) ◽

pp. 3369-3386 ◽

Cited By ~ 27

Author(s):

Alberto Carrassi ◽

Stéphane Vannitsem

Keyword(s):

Data Assimilation ◽

Deterministic Model ◽

Model Error ◽

Variational Data Assimilation ◽

Noise Model ◽

Actual State ◽

Strong Constraint ◽

Variational Assimilation ◽

Uncorrelated Noise ◽

Practical Applications

Abstract In data assimilation, observations are combined with the dynamics to get an estimate of the actual state of a natural system. The knowledge of the dynamics, under the form of a model, is unavoidably incomplete and model error affects the prediction accuracy together with the error in the initial condition. The variational assimilation theory provides a framework to deal with model error along with the uncertainties coming from other sources entering the state estimation. Nevertheless, even if the problem is formulated as Gaussian, accounting for model error requires the estimation of its covariances and correlations, which are difficult to estimate in practice, in particular because of the large system dimension and the lack of enough observations. Model error has been therefore either neglected or assumed to be an uncorrelated noise. In the present work, an approach to account for a deterministic model error in the variational assimilation is presented. Equations for its correlations are first derived along with an approximation suitable for practical applications. Based on these considerations, a new four-dimensional variational data assimilation (4DVar) weak-constraint algorithm is formulated and tested in the context of a linear unstable system and of the three-component Lorenz model, which has chaotic dynamics. The results demonstrate that this approach is superior in skill to both the strong-constraint and a weak-constraint variational assimilation that employs the uncorrelated noise model error assumption.

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Optimal solution error covariance in highly nonlinear problems of variational data assimilation

Nonlinear Processes in Geophysics ◽

10.5194/npg-19-177-2012 ◽

2012 ◽

Vol 19 (2) ◽

pp. 177-184 ◽

Cited By ~ 8

Author(s):

V. Shutyaev ◽

I. Gejadze ◽

G. J. M. Copeland ◽

F.-X. Le Dimet

Keyword(s):

Nonlinear Dynamics ◽

Data Assimilation ◽

Optimal Solution ◽

Nonlinear Problems ◽

Variational Data Assimilation ◽

Model Parameters ◽

Efficient Computation ◽

Viscous Term ◽

Highly Nonlinear ◽

The Cost

Abstract. The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can be approximated by the inverse Hessian of the cost function. For problems with strongly nonlinear dynamics, a new statistical method based on the computation of a sample of inverse Hessians is suggested. This method relies on the efficient computation of the inverse Hessian by means of iterative methods (Lanczos and quasi-Newton BFGS) with preconditioning. Numerical examples are presented for the model governed by the Burgers equation with a nonlinear viscous term.

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