Sensitivity of response functions in variational data assimilation for joint parameter and initial state estimation

Abstract

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find simultaneously unknown parameters and initial state of the model. A response function is considered as a functional of the optimal solution after assimilation. The sensitivity of the response function to the observation data is studied. The gradient of the response function with respect to observations is related to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. Based on the Hessian of the original cost function, the solvability of the non-standard problem is studied. An algorithm to compute the gradient of the response function with respect to observation data is formulated and justified. A numerical example is presented for variational data assimilation problem for the Baltic Sea thermodynamics model.