Singular vectors and estimates of the analysis-error covariance metric

An important ingredient of ensemble forecasting is the computation of initial perturbations. Various techniques exist to generate initial perturbations. All these aim to produce an ensemble that at initial time, reflects the uncertainty in the initial condition. In this paper a method for computing singular Vectors consistent with current estimates of the analysis-error statistics is proposed and studied. The singular-vector computation is constrained at initial time by the Hessian of the three-dimensional variational assimilation (3D-Var) cost function in a way which is consistent with the operational analysis procedure. The Hessian is affected by the approximations made in the implementation of 3D-Var; however, it provides a more objective representation of the analysis-error covariances than other metrics previously used to constrain singular vectors. Experiments are performed with a T21L5 Primitive-Equation model. To compute the singular vectors we solve a generalized eigenvalue problem using a recently developed algorithm. it is shown that use of the Hessian of the cost function can significantly influence such properties of singular vectors as horizontal location, vertical structure and growth rate. The impact of using statistics of observational errors is clearly visible in that the amplitude of the singular vectors reduces in data-rich areas. Finally the use of an approximation to the Hessian is discussed.