Multistep and Multi-scale Variational Data Assimilation: Spatial Variations of Analysis Error Variance

Spectral formulations were developed recently (Xu et al. 2016, Tellus A, 68, 26 pages) to address several important issues in multistep and multi-scale variational data assimilation. However, the analysis error variance computed from these spectral formulations is constant and thus limited to represent only the spatially averaged error variance. When the coarse-resolution observations used in the first step of multi-step and multi-scale variational data assimilation become increasingly non-uniform and/or sparse, the error variance of the first-step analysis tends to have increasingly large spatial variations. In this case, it is necessary to overcome the limitation caused by the constant analysis error variance estimated from the spectral formulations. To this end, semi-empirical 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 non-uniformly distributed observations without periodic extension). Its estimated analysis error variance closely captures the spatial variation of the benchmark truth. Then, three different formulations are constructed for using the estimated analysis error variance to modify the analysis error covariance computed from the spectral formulations. The first formulation is conventional in which the covariance is modulated by the spatially varying standard deviation separately via each entry of the covariance to retain the self-adjointness. The second and third formulations are new, in which the modulation is realigned and broadened to capture the banded structure revealed by the benchmark truth and yet still retain the self-adjointness. The successively improved accuracies of these three formulations and their increasingly positive impacts on the two-step variational analysis (or multi-step variational analysis in first two steps) are demonstrated by idealized experiments.