Dynamical effects of inflation in ensemble-based data assimilation under the presence of model error

Abstract

Covariance inflation is one of the necessary tools enabling the success of ensemble Kalman filters (EnKFs) in high-dimensional spaces and in the presence of model error. Inflation maintains the ensemble variance to a sufficiently large value, counteracting the variance damping at analysis times and its underestimation arising from model and sampling errors. In this work, we investigate the effect of inflation on the dynamics of the EnKF ensemble. When the focus is on the recursive full cycle forecast–analysis–forecast, an apparently counterintuitive effect of multiplicative inflation appears in the span of the ensemble in the EnKF. In particular, we demostrate that multiplicative inflation changes the alignment of ensemble anomalies on to weakly stable backward Lyapunov vectors. Whereas the ensemble is expected to collapse on to the subspace corresponding to the unstable portions of the Lyapunov spectrum, the use of multiplicative inflation contributes to the retention of anomalies beyond that subspace. Given that the presence of model error implies that the analysis error is no longer fully confined in the local unstable subspace, this feature of multiplicative inflation is of paramount importance for optimal filtering. We propose hybrid schemes, whereby additive perturbations complement multiplicative inflation by suitably increasing the dimension of the subspace spanned by the ensemble. The use of hybrid schemes improves analysis root-mean-squared error in the Lorenz 96 model compared with the use of multiplicative inflation alone, emphasizing the role of model dynamics when designing additive inflation schemes.