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Sensitivity of functionals with respect to observations in variational data assimilation

Abstract The problem of variational data assimilation for a model of ocean thermodynamics is formulated as an optimal control problem to find the boundary heat flux. The sensitivity of functionals of the optimal solution with respect to observations is studied. Computing the gradient of the functionals is reduced to the solution of a non-standard problem which is a coupled system involving direct and adjoint equations with mutually dependent boundary values. Solvability of the non-standard problem is studied based on the Hessian of the original cost function. An algorithm for computing the gradient of the response function related to the mean surface temperature is developed and justified. Numerical examples are presented for the Baltic Sea thermodynamics model.