Sensitivity of functionals in problems of variational assimilation of observational data

Abstract

The problem of the variational assimilation of observational data is stated for a nonlinear evolution model as a problem of optimal control in order to find the function of initial condition. The operator of the model, and consequently the optimal solution, depend on parameters that may contain uncertainties. A functional of the solution of the problem of variational data assimilation is considered. Using the method of second-order adjoint equations, the sensitivity of the functional in respect to the model parameters is studied. The gradient of the functional is expressed through solving a “nonstandard” (nonclassical) problem that involves the coupled system of direct and adjoint equations. The solvability of the nonstandard problem using the Hessian of the initial functional of observations is studied. Numerical algorithms for solving the problem and computing the gradient of the functional under consideration are developed with respect to the parameters. The results of the studies are applied in the problem of variational data assimilation for a 3D ocean thermodynamic model.