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

Shutyaev, V. P. and Gejadze, I. Y. (2011) Adjoint to the Hessian derivative and error covariances in variational data assimilation. Russian Journal of Numerical Analysis and Mathematical Modelling, 26 (2). pp. 179-188. ISSN 0927-6467

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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.