This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data‐assimilation applications. A self‐contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical‐analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross‐covariance functions on three‐dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross‐covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross‐covariance functions on S2, referred to as being space‐limited, cannot be obtained using truncated spectral expansions. Compactly supported and space‐limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data‐analysis algorithms. Convolution integrals leading to practical examples of compactly supported covariance and cross‐covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O < di,dj ≦, where |. | denotes Euclidean distance in R3. the parameters di and dj are ‘cut‐off’ distances. Closed‐form expressions are determined for classes of convolution cross‐covariance functions Cij(x,y) := (gi * gj)(x‐y), i ≠ j, and convolution covariance functions Cii(x,y) := (gi * gi)(x‐y), vanishing for |x ‐ y| > di + dj and |x ‐ y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second‐ and third‐order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x ‐ y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th‐order piecewise rational functions, are also constructed. These functions are used to develop space‐limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

Representation, representativity, representativeness error, forward interpolation error, forward model error, observation-operator error, aggregation error and sampling error are all terms used to refer to components of observation error in the context of data assimilation. This article is an attempt to consolidate the terminology that has been used in the earth sciences literature and was suggested at a European Space Agency workshop held in Reading in April 2014. We review the state of the art and, through examples, motivate the terminology. In addition to a theoretical framework, examples from application areas of satellite data assimilation, ocean reanalysis and atmospheric chemistry data assimilation are provided. Diagnosing representation-error statistics as well as their use in state-of-the-art data assimilation systems is discussed within a consistent framework.

Conventional optimal interpolation (OI) analysis systems solve the standard statistical analysis equations approximately, by invoking a local approximation and a data selection procedure. Although solution of the analysis equations is essentially exact in the recent generation of global spectral variational analysis systems, these new systems also include substantial changes in error covariance modeling, making it difficult to discern whether improvements in analysis and forecast quality are due to exact, global solution of the analysis equations, or to changes in error covariance modeling.The formulation and implementation of a new type of global analysis system at the Data Assimilation Office, termed the Physical-space Statistical Analysis System (PSAS), is described in this article. Since this system operates directly in physical space, it is capable of employing error covariance models identical to those of the predecessor OI system, as well as more advanced models. To focus strictly on the effect of global versus local solution of the analysis equations, a comparison between PSAS and OI analyses is carried out with both systems using identical error covariance models and identical data. Spectral decomposition of the analysis increments reveals that, relative to the PSAS increments, the OI increments have too little power at large horizontal scales and excessive power at small horizontal scales. The OI increments also display an unrealistically large ratio of divergence to vorticity. Dynamical imbalances in the OI-analyzed state can therefore be attributed in part to the approximate local method of solution, and are not entirely due to the simple geostrophic constraint built into the forecast error covariance model. Root-mean-square observation minus 6-h forecast errors in the zonal wind component are substantially smaller for the PSAS system than for the OI system.

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This paper considers the incorporation of constraints to enforce physically based conservation laws in the ensemble Kalman filter. In particular, constraints are used to ensure that the ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. In certain situations filtering algorithms such as the ensemble Kalman filter (EnKF) and ensemble transform Kalman filter (ETKF) yield updated ensembles that conserve mass but are negative, even though the actual states must be nonnegative. In such situations if negative values are set to zero, or a log transform is introduced, the total mass will not be conserved. In this study, mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate nonnegativity constraints. Simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. In two examples, an update that includes a nonnegativity constraint is able to properly describe the transport of a sharp feature (e.g., a triangle or cone). A number of implementation questions still need to be addressed, particularly the need to develop a computationally efficient quadratic programming update for large ensemble.