The simplest nontrivial covariance models for use in 3D or 4D variational assimilation tend to be based on the horizontally isotropic Gaussian form. The reason for this is that this particular shape, or approximations to it, can be efficiently synthesized numerically by a variety of convenient methods. However, studies of the actual structure of forecast errors indicate that this simple choice is deficient, not only in its restriction to an isotropic form, but also because the Gaussian shape itself gives insufficient weight to both the smallest and the largest resolved scales in comparison to those intermediate scales close to the Gaussian's defining scale parameter.

We address these deficiencies by presenting a broader parametric family of distributions, of which the Gaussians are special members, but which also accommodates a very general tensorial prescription of anisotropy, together with adaptive control over the degree of spatial "kurtosis" of the shape of the covariance distribution. Our new family of covariance models enjoys several algebraically convenient attributes, including the fact that the set of shapes of the implied power-spectra are exactly of the kind that are accommodated by the same parameteric model applied in the Fourier domain. Because each member of the proposed family can be formed as an additive mixture of anisotropic Gaussians, the same efficient numerical methods that facilitate the practical application to variational assimilation of these simpler covariances can be extended without difficulty to the application of these more appropriate and more general covariance forms.

Consideration is given both to the practical implementation of adaptive approximations of these covariances to variational assimilation, and to the objective estimation of the underlying covariance parameters, either from observational data or from judiciously chosen diagnostics obtained from a forecast ensemble.