Numerical Aspects of the Application of Recursive Filters to Variational Statistical Analysis. Part II: Spatially Inhomogeneous and Anisotropic General Covariances

Abstract

In this second part of a two-part study of recursive filter techniques applied to the synthesis of covariances in a variational analysis, methods by which non-Gaussian shapes and spatial inhomogeneities and anisotropies for the covariances may be introduced in a well-controlled way are examined. These methods permit an analysis scheme to possess covariance structures with adaptive variations of amplitude, scale, profile shape, and degrees of local anisotropy, all as functions of geographical location and altitude. First, it is shown how a wider and more useful variety of covariance shapes than just the Gaussian may be obtained by the positive superposition of Gaussian components of different scales, or by further combinations of these operators with the application of Laplacian operators in order for the products to possess negative sidelobes in their radial profiles. Then it is shown how the techniques of recursive filters may be generalized to admit the construction of covariances whose characteristic scales relative to the grid become adaptive to geographical location, while preserving the necessary properties of self-adjointness and positivity. Special attention is paid to the problems of amplitude control for these spatially inhomogeneous filters and an estimate for the kernel amplitude is proposed based upon an asymptotic analysis of the problem. Finally, a further generalization of the filters that enables fully anisotropic and geographically adaptive covariances to be constructed in a computationally efficient way is discussed.

Introduction of the GSI into the NCEP Global Data Assimilation System

“…The use of recursive filters for the application of the background error in GSI is the same as is outlined in WPP02 and Purser et al (2003a). Unlike in WPP02, however, where a single horizontal length scale was chosen, three sets of horizontal scales are applied in order to achieve the ''fat tailed'' feature in the spectrum of the error covariance.…”

Heterogeneous Convective-Scale Background Error Covariances with the Inclusion of Hydrometeor Variables

“…where S is a gridpoint variance scaling factor; U ih is the application of high order, homogeneous (Purser et al 2003a) or inhomogeneous (Purser et al 2003b) recursive filters to impose horizontal correlations; U y is the application of vertical correlations through homogeneous empirical orthogonal functions (EOF) and U p changes the control variables to model variables through a physical transform. The gridpoint variance scaling factor is not used in this study because background error covariances are taken to be piecewise constant over the geographical masks.…”

Impact of hybrid GSI analysis using ETR ensembles

“…In reality, the background-error covariances may vary substantially depending on the flow of the day (Wang et al 2007a). In order to address this issue, there were attempts to include some amount of spatial inhomogeneity and anisotropy in standard covariance models used in 3D Var (Wu et al 2002;Purser et al 2003). Ensemble based data assimilation in various forms of ensemble Kalman filter (EnKF) is an alternative to variational approaches.…”