Singular Vector Calculations with an Analysis Error Variance Metric

Singular vectors of the navy's global forecast model are calculated using an initial norm consistent with an estimate of analysis error variance provided by the Naval Research Laboratory's (NRL) Atmospheric Variational Data Assimilation System (NAVDAS). The variance estimate is based on a decomposition of the block diagonal preconditioner for the conjugate-gradient descent algorithm used in NAVDAS. Because the inverse square root of the operator that defines the variance norm is readily computed, the leading singular vectors are obtained using a standard Lanczos algorithm, as with diagonal norms such as total energy. The resulting singular vectors are consistent with the expected distribution of analysis errors. Compared with singular vectors based on a total energy norm, the variance singular vectors at initial time have less amplitude over well-observed areas, as well as greater amplitude in the middle and upper troposphere. The variance singular vectors are in some ways similar to the full covariance (Hessian) singular vectors developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). However, unlike the Hessian singular vectors, the variance singular vectors exhibit only minor difference in structure and growth rate compared with total energy singular vectors. This is because the variance singular vectors exclude covariance information used in NAVDAS that significantly penalizes smaller scales. The 20 leading analysis error variance singular vectors explain approximately the same fraction of forecast error variance as the total energy singular vectors in a linear context, but less in a nonlinear context. Deficiencies in the current experimental configuration are among the reasons suspected for this. Implications for targeted observing are also examined. The results show that the variance norm can have a significant impact on determining the locations for supplemental observations.