Determine whether forecast covariances in the two-layer quasi-geostrophic model exhibit inhomogeneous or anisotropic structure exploitable by GenGC

Ascertain whether the forecast covariance and correlation matrices arising in the two-layer quasi-geostrophic model used in this study exhibit notable inhomogeneity or anisotropy that can be leveraged by Generalized Gaspari–Cohn (GenGC) localization, thereby justifying its application; if such structure exists, characterize it and derive appropriate spatially varying GenGC hyperparameter fields.

Background

In the quasi-geostrophic (QG) experiments, the authors considered multiple localization methods but opted not to apply GenGC. While the climatological stream functions and covariances show inhomogeneous and anisotropic features, the authors report that this did not translate into significant inhomogeneity or anisotropy in the forecast covariances of their large-ensemble DA experiment.

This raises an unresolved question: whether, under different regimes or analyses, the QG model’s forecast covariance structure contains exploitable inhomogeneity or anisotropy for GenGC localization, which would warrant its use and tuning in this context.

References

We do not apply localization with GenGC to the QG model, as we we were unable to discover notable inhomogeneous covariance/correlation structure one could leverage with GenGC. Indeed, while both climatological stream functions (Figure~\ref{fig:qg psi2 example}(b)) and climatological covariances (not shown) exhibit inhomogeneous and anisotropic features, for example the clear zonal structure in the stream function, this did not translate into significant inhomogeneity or anisotropy in the forecast covariance of our large ensemble DA experiment.

— Numerical study of high-dimensional covariance estimation and localization for data assimilation  (2508.18299 - Gilpin et al., 22 Aug 2025) in Section 4.2, Localization implementation