Observation targeting with a second-order adjoint method for increased predictability

The efficiency of current adjoint-based observations targeting strategies in variational data assimilation is closely determined by the underlying assumption of a linear propagation of initial condition errors into the model forecasts. A novel targeting strategy is proposed in the context of four-dimensional variational data assimilation (4D-Var) to account for nonlinear error growth as the forecast lead time increases. A quadratic error growth model is shown to maintain the accuracy in tracking the nonlinear evolution of initial condition perturbations, as compared to the first-order approximation. A second-order adjoint model is used to provide the derivative information that is necessary in the higher-order Taylor series approximation. The observation targeting approach relies on the dominant eigenvectors of the Hessian matrix associated with a specific forecast error aspect as an indicator of the directions of largest quadratic error growth. A comparative qualitative analysis between observation targeting based on first- and second-order adjoint information is presented in idealized 4D-Var experiments with a two-dimensional global shallow-water model. The results indicate that accounting for the quadratic error growth in the targeting strategy is of particular benefit as the forecast lead time increases.