Optimization of parameters of the Multigrid Beta Filter (MGBF) based on a covariance diagnostics approach

Performance of the Multigrid Beta Filter (MGBF), a new technique for synthesis of covariances introduced at the Environmental Modeling Center (EMC), critically depends on the choice of its parameters. Aspect tensors control the spread and shape of covariances at each grid generation of the multigrid structure used for application of the filter, and the scale weights define contributions of grid generations to the resulting covariance. In the previous stages of effort to estimate these parameters, we first used error statistics derived by the NMC method in a series of simulations of a standalone version of MGBF, and then we refined these results through application of a machine learning method to results of the Rapid Refresh Forecast System (RRFS) version of Gridpoint Statistical Interpolation (GSI) running with MGBF and using GSI with the recursive filter as a reference.

In this presentation we describe an attempt to give a final polish to the MGBF performance through an iterative improvement based on diagnostics derived from the GSI runs with MGBF, following the approach used by Hollingsworth and Lönnberg (1986, 1989), and Desroziers et al. (2001, 2005).

Assuming that observation and background errors are independent, and that the derived analysis is optimal, we can get a sample of error covariances that can be used to find the most optimal set of scale weight for the derived analysis. Technically, a generic sample covariance, counted as a function of the radial distance of separation between its two observation points and discretized into a set of histogram bins, can be fitted in minimum least-squares sense with a MGBF approximated using a weighted sum of self-convolved Gaussians, to look for the best possible scale weights. The successive use and corrections of the derived weights in the next analysis cycles will allow for the improvement of these estimates, provided that the true statistical behavior remains uniform.