Development of Global Versions of the Multigrid Beta Filter (MGBF)
Authors: Rancic Miodrag , Purser R. James , De Pondeca Manuel , Lei Ting 
Published: March 2024

The Multigrid Beta Filter (MGBF) is a technique for the synthesis of background error covariances under development at the National Centers for Environmental Prediction (NCEP).&#160; High computational efficiency in a massively parallel computing environment is achieved through the application of a compact-support quasi-Gaussian response function formed by the polynomial profile of the beta density distribution and the multigrid strategy for extension of the spatial coverage.&#160; The paper will discuss strategies for applying the MGBF on the global domain, based on the gnomonic cubed-sphere grid used by the present operational Unified Forecast System's (UFS) FV3 model, but also with the option to accommodate less structured global analysis grids possessing spatial resolutions which might vary smoothly in the horizontal.&#160; A preliminary version of the global MGBF had been developed using an &#8220;equiangular cubed&#8221; sphere for filtering, which requires remapping between the analysis and the filter grid, and special treatment of areas close to edges and corners of the cubed sphere.&#160; It is based on the application of the &#8220;radial&#8221; version of the MGBF, whose horizontal response stencil over its elliptical support footprint is computed in a single step.&#160; Updates of this method include transition to the related &#8220;duo-grid&#8221;, recently recommended for reduction of edge/corner noise in FV3, which will reduce the time for remapping between the analysis and the filter grid, and application of a generalized version of the radial MGB filter which should fully eliminate the need for grid overlapping.&#160; As an alternative direction of development, we consider the semi-structured global &#8220;Fibonacci grid&#8221; configuration, adapted by way of small polar overset regions to remove their polar singularities.&#160; Filtering on such a grid could be based on application of the generalized radial filter, but also can use a &#8220;line&#8221; filter, whose horizontal response is built up in a sequence of successively parallel convolution sweeps of the one-dimensional filter along generalized lines of the grid resembling how NCEP&#8217;s present operational Recursive Filter covariance scheme synthesizes covariances. The line filters enjoy a computational efficiency advantage over the radial filters owing to the smaller number of floating-point operations involved, if operating on a moderate number of processing elements.&#160; Although the Fibonacci grid option incurs an additional expense of requiring interpolations between the filtering grid and the final FV3 grid of the analysis increments, it possesses the further potential advantage of being able to incorporate a smooth spatial variation of resolution, which makes it an attractive option for any future versions of the UFS that are constructed to have a horizontally adaptive, or unstructured, grid.&#160; These points will be discussed in our presentation, the progress towards the implementation of these two approaches to generalizing the MGBF will be illustrated, and their comparative performance evaluated.