In this study, a global three-dimensional variational analysis system is formulated in model grid space. This formulation allows greater flexibility (e.g. inhomogeneity and anisotropy) for background error statistics. A simpler formulation, inhomogeneous only in the latitude direction, was chosen for these initial tests. The background error statistics are defined as functions of the latitudinal grid and are estimated with the NMC method. The horizontal scales of the variables are obtained through the variances of the variables and of their Laplacian. The vertical scales are estimated through the statistics of the vertical correlation of each variable and are applied locally using recursive filters. For the multivariate correlation between wind and mass fields, a statistical linear relationship between the stream function and the balanced part of temperature and surface pressure is assumed. A localized correlation between the velocity potential and the stream function is also used to account for the positive correlation between the vorticity and divergence in the planetary boundary layer. Horizontally, the global domain is divided into three pieces so that efficient spatial recursive filters can be used to spread out the information from the observation locations. This analysis system is tested against the operational Spectral Statistical-Interpolation analysis system at the National Centers for Environmental Prediction. The results indicate that 3D Var in physical space is as effective as 3D Var in spectral space in the extratropics and yields superior results in the tropics as a result of the latitude dependence of the background error statistics.

SUMMARYRecent years have seen a resurgence of interest in a variety of non-standard computational grids for global numerical prediction. The motivation has been to reduce problems associated with the converging meridians and the polar singularities of conventional regular latitude-longitude grids. A further impetus has come from the adoption of massively parallel computers, for which it is necessary to distribute work equitably across the processors; this is more practicable for some non-standard grids. Desirable attributes of a grid for high-order spatial finite differencing are: (i) geometrical regularity; (ii) a homogeneous and approximately isotropic spatial resolution; (iii) a low proportion of the grid points where the numerical procedures require special customization (such as near coordinate singularities or grid edges); (iv) ease of parallelization.One family of grid arrangements which, to our knowledge, has never before been applied to numerical weather prediction, but which appears to offer several technical advantages, are what we shall refer to as 'Fibonacci grids'. These grids possess virtually uniform and isotropic resolution, with an equal area for each grid point. There are only two compact singular regions on a sphere that require customized numerics. We demonstrate the practicality of this type of grid in shallow-water simulations, and discuss the prospects for efficiently using these frameworks in three-dimensional weather prediction or climate models.

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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.

A recursive filter objective analysis method is described. It is a "successive approximation" system with the particular feature of locally varying scaling, making it especially appropriate for dealing with inhomogeneous data. Attention is given to proper treatment of lateral boundaries, which permit its use in limited domains. Thissystem provides estimates of input data quality that can be used for editing datasets before their distribution and for the weighting of data in application by other users, Two- and three-dimensional versions of the analysis operating on a Cartesian grid are used operationally at the National Environmental Satellite and Data InformationService. They are used both in the production of data and for quality control prior to dissemination. Examplesof these applications are given.