A comparison of prescriptions for using background field diagnostics to adapt covariances to the ambient flow in a 3D Variational assimilation

A computationally less costly alternative to the method of 4D variational assimilation (4D Var) that also captures some dynamically adaptive features is to apply 3D Var, but with covariances for background error prescribed anisotropically as diagnostic functions of the background field itself. The computationally necessary properties of self-adjointness and non-negativity of any practical covariance are guaranteed by synthesizing it as a product of a pair of mutually adjoint "square-root" factors. These factors themselves are then not compelled to conform to any restrictive symmetries, but can be built up using both additive and multiplicative combinations of more basic spatial smoothing operators in order to exhibit desirable characteristic profile shapes, spatial patterns of anisotropy and degrees of coupling among different dependent variables.

At NCEP we are developing a systematic procedure to synthesize covariances in this fashion, based on the weighted superposition of a small number of generally anisotropic quasi-Gaussian filters that each mimic the outcome of a diffusive process. The spatial spread of a quasi-Gaussian response at each location is quantified by its "aspect tensor", defined as twice the product of the pretended "diffusivity" and the "duration" over which it must act to produce that response. The geometrical "Hexad" algorithm and its variants (described elsewhere at this conference) provide analytic tools facilitating efficient construction, on any smooth computational lattice, of quasi-Gaussian filters with arbitrarily prescribed aspect tensors. But the aspect tensors themselves first must be defined somehow at every location, preferably in a way that enables information implicit in the shapes of the distributions of the dynamical variables of the background field to be extracted and intelligently interpreted to suggest the background error covariance anisotropies (and hence, the anisotropies of the analysis increments) that are, on average, most likely to benefit the quality of the analysis that results.

Here, we examine three different strategies suggested to accomplish this objective. The first is essentially the method of Riishojgaard, by which the covariance is made dependent upon an effective relative displacement that resembles a Euclidean distance but includes among its components the additional terms coming from the differences in the values those predominantly advected quantities of the background. (These are expected to exhibit patterns of anisotropy that correlate with those that might be beneficially imposed upon the new increments.) The second is a localized modification of the semi- geostropic transformation method suggested by Desroziers, by which analysis influence across strong frontal boundaries is inhibited. The third appoach we examine is approximately equivalent to an assumption that the covariance is simply a passive kinematic distortion, accumulated over a predetermined period of a few hours, of what would initially have been a covariance possessing the traditional attributes of horizontal isotropy and homogeneity.

We review and compare the behavior of these methods, and minor variants of them, according to our accumulating experience applying them to real data within the Mesoscale Eta Data Assimilation System.