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This paper presents a practical computational approach to quantify the effectof individual observations in estimating the state of a system. Such ananalysis can be used for pruning redundant measurements, and for designingfuture sensor networks. The mathematical approach is based on computing thesensitivity of the reanalysis (unconstrained optimization solution) withrespect to the data. The computational cost is dominated by the solution of alinear system, whose matrix is the Hessian of the cost function, and is onlyavailable in operator form. The right hand side is the gradient of a scalarcost function that quantifies the forecast error of the numerical model. Theuse of adjoint models to obtain the necessary first and second orderderivatives is discussed. We study various strategies to accelerate thecomputation, including matrix-free iterative solvers, preconditioners, and anin-house multigrid solver. Experiments are conducted on both a small-sizeshallow-water equations model, and on a large-scale numerical weatherprediction model, in order to illustrate the capabilities of the newmethodology.

Efficient methods for computing observation impact in 4D-Var data assimilation