# Adjoint to the Hessian derivative and error covariances in variational data assimilation

V. P. Shutyaev, I. Y. Gejadze

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems.
Language English 179-188 10 Russian Journal of Numerical Analysis and Mathematical Modelling 26 2 10.1515/RJNAMM.2011.010 Published - Apr 2011

### Fingerprint

Data Assimilation
Derivatives
Derivative
Optimal Solution
Covariance Operator
Error Analysis
Tangent line
Error analysis
Optimal Control Problem
Linear Model
Initial conditions
Exact Solution
Nonlinearity

### Keywords

• theoretical aspects
• hessian derivative
• error covariances
• variational data
• assimilation

### Cite this

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Adjoint to the Hessian derivative and error covariances in variational data assimilation. / Shutyaev, V. P.; Gejadze, I. Y.

In: Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 26, No. 2, 04.2011, p. 179-188.

Research output: Contribution to journalArticle

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N2 - The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems.

AB - The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems.

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