Objective filtering of the local correlation tensor

Abstract

The local correlation tensor (LCT), also referred to as the local correlation Hessian, the inverse of which is known as the Daley tensor, has proven a useful diagnostic of the spatial variability of background‐error correlations in data assimilation. The LCT (or its inverse) is also used in several correlation models, including those based on recursive filters, the diffusion equation and spatial deformations. It can be estimated from the variances of the background errors and of their spatial derivatives. Additional terms involving the spatial derivatives of the standard deviations are often neglected. This approximation is first discussed for ensembles of forecasts at the global scale.In the context of numerical weather prediction (NWP), only limited ensemble sizes are computationally affordable, meaning that the LCT is affected by sampling noise. The estimation of the LCT may be improved by an efficient spatial filtering designed to remove this sampling noise. Recently, a linear filtering theory was developed by the authors for the purpose of filtering background‐error covariances objectively, with applications to variance filtering and localization (in a convective‐scale model). We discuss several practical filters for the LCT that are designed to preserve its symmetric positive‐definite character. This can be achieved by filtering the raw LCT itself, or by filtering the numerator and the denominator separately within the aforementioned approximation. This also requires a positive filter, a feature not present in earlier attempts. Finally, it is shown that the LCT can be estimated robustly from small ensembles and that this estimation benefits from our objective spatial filtering, equivalent to increasing the sample size by a factor of 1.5–3 depending on the model variable and the vertical level at which it is defined.