Wavelet model of multivariate prior covariance

Initialization of numerical weather-prediction models uses data assimilation to specify an optimal analysis state derived from observations and the model background state. The associated covariance structures are crucial factors. The full background-error covariance (BEC) matrix would contain a number of entries of order the square of the nominal degrees of freedom, which for realistic models is prohibitive even to store, let alone compute with. To obtain the ideal diagonal BEC would require solving an intractably large eigenvector problem. A reasonable approach is to approximate the eigenvectors by appropriate known functions, then construct diagonal BEC values accordingly. Until now, the Weather Research and Forecasting (WRF) model data assimilation (WRFDA) used recursive filters for the horizontal factors of these functions. Here an option is introduced to replace those with a little-known wavelet type, the wavelet Parseval frame (i.e., left-invertible transform that preserves the sum of squares). The wavelet option has strong theoretical and empirical support in the reviewed literature. The pseudo-single observation tests presented here exhibit improved representation of horizontal spatial heterogeneity and anisotropy compared to recursive filters, comparable to the realism of the ensemble method, but at much less computational cost.