Heterogeneous Correlation Modeling Based on the Wavelet Diagonal Assumption and on the Diffusion Operator

Abstract

This article discusses several models for background error correlation matrices using the wavelet diagonal assumption and the diffusion operator. The most general properties of filtering local correlation functions, with wavelet formulations, are recalled. Two spherical wavelet transforms based on Legendre spectrum and a gridpoint spherical wavelet transform are compared. The latter belongs to the class of second-generation wavelets. In addition, a nonseparable formulation that merges the wavelets and the diffusion operator model is formally proposed. This hybrid formulation is illustrated in a simple two-dimensional framework. These three formulations are tested in a toy experiment on the sphere: a large ensemble of perturbed forecasts is used to simulate a true background error ensemble, which gives a reference. This ensemble is then applied to compute the required parameters for each model. A randomization method is utilized in order to diagnose these different models. In particular, their ability to represent the geographical variations of the local correlation functions is studied by diagnosis of the local length scale. The results from these experiments show that the spectrally based wavelet formulation filters the geographical variations of the local correlation length scale but it is less able to represent the anisotropy. The gridpoint-based wavelet formulation is also able to represent some parts of the geographical variations but it appears that the correlation functions are dependent on the grid. Finally, the formulation based on the diffusion represents quite well the local length scale.