Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation Advances in Meteorology

When the coarse-resolution observations used in the first step of multiscale and multistep variational data assimilation become increasingly nonuniform and/or sparse, the error variance of the first-step analysis tends to have increasingly large spatial variations. However, the analysis error variance computed from the previously developed spectral formulations is constant and thus limited to represent only the spatially averaged error variance. To overcome this limitation, analytic formulations are constructed to efficiently estimate the spatial variation of analysis error variance and associated spatial variation in analysis error covariance. First, a suite of formulations is constructed to efficiently estimate the error variance reduction produced by analyzing the coarse resolution observations in one- and two-dimensional spaces with increased complexity and generality (from uniformly distributed observations with periodic extension to nonuniformly distributed observations without periodic extension). Then, three different formulations are constructed for using the estimated analysis error variance tomodify the analysis error covariance computed from the spectral formulations. The successively improved accuracies of these three formulations and their increasingly positive impacts on the two-step variational analysis (ormultistep variational analysis in first two steps) are demonstrated by idealized experiments. Grant no. NA11OAR4320072 Grant no. N000141410281 Grant no. N000141712375