The Multigrid Beta Function Approach for Modeling of Background Error Covariance in the Real-Time Mesoscale Analysis (RTMA)

Abstract

Abstract We describe a method for the efficient generation of the covariance operators of a variational data assimilation scheme, which is suited to implementation on a massively parallel computer. The elementary components of this scheme are what we call “beta filters,” since they are based on the same spatial profiles possessed by the symmetric beta distributions of probability theory. These approximately Gaussian (bell-shaped) polynomials blend smoothly to zero at the ends of finite intervals, which makes them better suited to parallelization than the present quasi-Gaussian “recursive filters” used in operations at NCEP. These basic elements are further combined at a hierarchy of different spatial scales into an overall multigrid structure formulated to preserve the necessary self-adjoint attribute possessed by any valid covariance operator. This paper describes the underlying idea of the beta filter and discusses how generalized Helmholtz operators can be enlisted to weight the elementary contributions additively in such a way that the covariance operators may exhibit realistic negative sidelobes, which are not easily obtained through the recursive filter paradigm. The main focus of the paper is on the basic logistics of the multigrid structure by which more general covariance forms are synthesized from the basic quasi-Gaussian elements. We describe several ideas on how best to organize computation, which led us to a generalization of this structure that made it practical so that it can efficiently perform with any rectangular arrangement of processing elements. Some simple idealized examples of the applications of these ideas are given.