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Advances in Meteorology
Volume 2018 (2018), Article ID 7931964, 17 pages
Research Article

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

Qin Xu1 and Li Wei2

1NOAA/National Severe Storms Laboratory, Norman, OK, USA
2Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, OK, USA

Correspondence should be addressed to Qin Xu; vog.aaon@ux.niq

Received 24 April 2017; Revised 24 November 2017; Accepted 3 December 2017; Published 7 February 2018

Academic Editor: Shaoqing Zhang

Copyright © 2018 Qin Xu and Li Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 to modify 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 (or multistep variational analysis in first two steps) are demonstrated by idealized experiments.