Prospects for Using a Multigrid Filter to Characterize the Errors of a Variational Analysis

When a variational analysis is performed, it is generally desirable to know how accurate it is at each location. Although the variational formulation itself provides a formal statistical answer in principle, the result cannot be obtained exactly, since computing it would involve inversions of matrices of impractically-large orders. However, if we allow the approximating assumption that, at each point, the contributing observational data of each type can be almost equivalently represented as a smoothed-out continuum possessing the same precision-weighted spatial density as the actual data, then it follows that a local representation of the spectral profile of analysis error at each location becomes feasible. At EMC we have developed a new “MultiGrid Beta Filter” (MGBF) technique for the synthesis of background error covariances for variational analysis. The quasi-Gaussian beta filters can serve as smoothing
operators to produce the needed precision-weighted data densities. Moreover, since the spectral response of each beta filter contributing to a generic multigrid synthesis is known, it also becomes possible to exploit the MGBF machinery to find an approximating local projection into the space spanned
by beta filter covariance representations of the analysis error’s estimated spectral profile. Thus, we can adaptively represent the analysis error covariance itself by an approximating MGBF with filtering amplitudes at each scale derived objectively and automatically. We shall describe the details of this
method, and the preliminary assessment of how well this approach performs relative to other methods of estimating the analysis error