Development of an Adaptive Covariance Operator in a Multigrid Framework Using Filters Based on Beta Distributions, with Applications in a Three-Dimensional Real-Time Mesoscale Analaysis

A new method of generating background error covariance operators in a massively-parallel computer architecture is described. It uses a synthesis based on a combination of Beta-distribution filters with a multigrid algorithmic structure. This is intended for use in an enhancement of NOAA’s operational Real-Time Mesoscale Analysis (RTMA). The current two-dimensional RTMA of surface and near-surface data presently employs an anisotropic version of the Gridpoint Statistical interpolation’s background error covariance, which is synthesized from a sequence of quasi-Gaussian numerical line-filters of a recursive kind. Owing to their recursive nature, these filters do not have a finite support, and consequently become progressively harder to parallelize efficiently as the technology advances towards ever greater numbers of available processors, since this is always accompanied by a corresponding progression to ever-increasing spatial resolution. As the intended transition is made to a fully three-dimensional RTMA, these difficulties are only exacerbated. This has led to an examination of the alternative non-recursive quasi-Gaussian filters, based on Beta distributions, combined with a new synthesis of these elementary components that exploits the powerful ‘multigrid’ algorithmic paradigm.
The new ‘beta filters’ each compose a local fragment of the generalized square-root of the final covariance on a grid of resolution and subdomain size both commensurate with the scale of this quasi-Gaussian contribution. Since the reach of this filter is finite and involves only a modest number of target points, even in the full dimensionality of the grid, it can be computed directly. Moreover, it can be carried out in parallel with similar operations applied to domains that, not only collectively cover the full domain at the finest grid resolution (using many processors), but also at several progressively coarser (by factors of powers of two) resolutions. In this way the total superposition involves many scales of quasi-Gaussian components weighted differently (and allowing geographical modulation, in general) in such a way that we are enabled to synthesize a very wide range of generalized square-root covariance operators. The final covariance is obtained by compounding the above superposition synthesis by exactly the adjoint sequence (in reverse order) of multigrid filtering operators, which guarantees self-adjointness and positivity of the whole covariance.
We shall describe the Beta function filters and the multigrid structure in detail, and show preliminary results. We shall also discuss the prospect of using the multigrid structure to incorporate elements of nonlinear dynamical balance in an appropriately scale-dependent way, and the opportunities to incorporate dynamically adaptive spatial stretching and cross-variable correlations by exploiting localized (value and derivative) diagnostics taken from forecast ensemble member sample correlations when these are available. Finally, we describe steps to be taken to incorporate this proposed covariance operator as a modular component of the Joint Effort for Data assimilation Integration (JEDI) assimilation framework.