Construction and application of covariance functions with variable length-fields

This article focuses on construction, directly in physical space, of three-dimensional covariance functions parametrized by a length-field, and on an application of these functions to improve the representation of the Quasi-Biennial Oscillation (QBO) in the Goddard Earth Observing System, Version 4 (GEOS-4) data assimilation system. The covariance functions are obtained by fusing collections of auto-covariance functions having different constant length-scales with their associated cross-covariance functions. This construction yields covariance functions with length-scales that can vary arbitrarily over any finite partition of the spatial domain. A simple, and also motivating application of these functions is to the case where the length-scale varies in the vertical direction only. The class of covariance functions with variable length-fields constructed in this article will be called multi-level to associate them with this application. The multi-level covariance functions extend well-known single-level covariance functions depending only on a constant length-scale. Generalizations of the familiar first- and third-order autoregressive covariances in three dimensions are given, providing multi-level covariances with zero and four continuous derivatives at zero separation, respectively. Multi-level piecewise rational covariances with two continuous derivatives at zero separation are also provided. Multi-level power-law covariances are constructed with continuous derivatives of all orders. Additional multi-level covariance functions are constructed using the Schur product of single- and multi-level covariance functions. A multi-variate, multi-level power-law covariance with a large troposphere-to-stratosphere length-field gradient is employed to reproduce the QBO from sparse radiosonde wind observations in the tropical lower stratosphere. This covariance model is described along with details of the assimilation experiments. The new covariance model is shown to represent the vertical wind shear associated with the QBO much more effectively than the multi-variate, multi-level covariance model in the baseline GEOS-4 system.