Multilevel Monte Carlo methods for ensemble variational data assimilation

Abstract. Ensemble variational data assimilation relies on ensembles of forecasts to estimate the background error covariance matrix B. The ensemble can be provided by an ensemble of data assimilations (EDA), which runs independent perturbed data assimilation and forecast steps. The accuracy of the ensemble estimator of B is strongly limited by the small ensemble size that is needed to keep the EDA computationally affordable. Here we investigate the potential of the multilevel Monte Carlo (MLMC) method, a type of multifidelity Monte Carlo method, to improve the accuracy of the standard Monte Carlo estimator of B while keeping the computational cost of ensemble generation comparable. MLMC exploits the availability of a range of discretization grids, thus shifting part of the computational work from the original assimilation grid to coarser ones. MLMC differs from the mere averaging of statistical estimators, as it ensures that no bias from the coarse-resolution grids is introduced in the estimation. The implications for ensemble variational data assimilation systems based on EDAs are discussed. Numerical experiments with a quasi-geostrophic model demonstrate the potential of the approach, as MLMC yields more accurate background error covariances and reduced analysis error. The challenges involved in cycling a multilevel variational data assimilation system are identified and discussed.