A Novel Localized Fast Multipole Method for Computations With Spatially Correlated Observation Error Statistics in Data Assimilation

Abstract

Several observation types (e.g., geostationary satellite and Doppler radar observations) have recently been found to exhibit strong spatial error correlations. Including these error statistics in data assimilation for numerical weather prediction can improve analysis quality and forecast skill. Moreover, it allows for increases in the spatial density of observations assimilated, which is needed for the provision of information on appropriate scales for high‐resolution forecasting. However, introducing correlated error statistics may increase the computational complexity and parallel communication costs of matrix‐vector products involving observation precision matrices (inverse observation error covariance matrices). Without new approaches, we cannot take full advantage of new observation uncertainty estimates. We develop a new numerical approximation method based on a particular type of fast multipole method and a domain localization approach. The basic idea is to divide the observation domain into boxes and then separate calculations of matrix‐vector products according to the partition. These calculations can be done in parallel with very low communication overheads. The new method is easy to implement and parallelize, and it is applicable to a wide variety of observation precision matrices. We applied the new method to a simple variational data assimilation problem and found that the computational cost of the variational minimization was dramatically reduced while preserving analysis accuracy across a range of scales. The new method has the potential to be used as an efficient technique for practical applications where a large number of observations with mutual error correlations need to be assimilated quickly.