This chapter presents the mathematical framework to evaluate the sensitivity
of a model forecast aspect to the input parameters of a nonlinear four-dimensional
variational data assimilation system (4D-Var DAS): observations, prior state
(background) estimate, and the error covariance specification. A fundamental
relationship is established between the forecast sensitivity with respect to the
information vector and the sensitivity with respect to the DAS representation of the
information error covariance. Adjoint modeling is used to obtain first- and second-order
derivative information and a reduced-order approach is formulated to alleviate the
computational cost associated with the sensitivity estimation. Numerical results from
idealized 4D-Var experiments performed with a global shallow water model are used to
illustrate the theoretical concepts.
Keywords: Estimation theory, atmospheric data assimilation, error statistics,
sensitivity analysis, adjoint model, large-scale optimization, Hessian matrix,
second-order derivative information, order reduction, variational data
assimilation, background estimate, error covariance, DAS representation, shallow
water model, error statistics, information-redundancy, suboptimal weighting, a
priori estimates, data thinning, super-obbing.
