A Second Order Adjoint Method to Targeted Observations

The role of the second order adjoint in targeting strategies is studied and analyzed. Most targeting strategies use the first order adjoint to identify regions where additional information is of potential benefit to a data assimilation system. The first order adjoint posses a restriction on the targeting time for which the linear approximation accurately tracks the evolution of perturbation. Using second order adjoint information it is possible to maintain some accuracy for longer time intervals, which can lead to an increase on the target time. We propose the use of the dominant eigenvectors of the Hessian matrix as an indicator of the directions of maximal error growth for a given targeting functional. These vectors are a natural choice to be included in the targeting strategies given their mathematical properties.