A Novel Numerical Approximation Method for Computations with Spatially Correlated Observation Error Statistics in Data Assimilation

Convection-permitting data assimilation requires observations with high spatial density and high temporal frequency to provide information on appropriate scales for high resolution forecasting. Those observation types (e.g., geostationary satellite data) were found to exhibit strong spatial error correlations. Explicitly introducing correlated error statistics in the assimilation may increase the computational complexity and parallel communication costs of the matrix-vector multiplications with the observation precision matrices (the inverse observation error covariance matrices). Therefore, without suitable approaches we cannot take full advantage of the new observation uncertainty estimates. In this work, we present a new numerical approximation method, called the local SVD-FMM, which is developed based on a particular type of the fast multipole method (FMM) using a singular value decomposition (SVD), and a domain localization approach. The basic idea of the local SVD-FMM is to divide the observation domain into boxes of (approximately) equal size and then separates the calculations of the matrix-vector products according to the domain partition. These calculations can be done in parallel with very low communication overheads. Moreover, the local SVD-FMM is easy to implement and applicable to a wide variety of the precision matrices. We applied the local SVD-FMM in a simple variational data assimilation system and found that the computational cost of the variational minimisation was dramatically reduced while preserving the accuracy of the analysis. This new method has the potential to be used as an efficient technique for practical data assimilation applications where a large volume of observations with mutual error correlations needs to be assimilated in a short period of time.