Correlation operators based on an implicitly formulated diffusion equation solved with the Chebyshev iteration

Correlation operators are used in the formulation of background-error covariance models in variational data assimilation (VDA) and for localizing low-rank sample estimates of background-error covariance matrices in ensemble VDA. This article describes new approaches for defining correlation operators based on diffusion operators. The starting point is a two-dimensional (2D) implicitly formulated diffusion operator on the sphere, which has been shown in previous works to support symmetric and positive-definite smoothing kernels that are closely related to those from the Matern correlation family. Different methods are proposed for solving the 2D implicit diffusion problem and these are compared with respect to their efficiency, accuracy, memory cost, ease of implementation and parallelization properties on high-performance computers. The methods described in this article are evaluated in a global ocean VDA system. An iterative algorithm based on the Chebyshev iteration, which uses a fixed number of iterations and pre-computed eigenvalue bounds, is shown to be particularly promising. Techniques for improving the parallelization aspects of the algorithm further are discussed.