Impact of non-stationarity on hybrid ensemble filters: A study with a doubly stochastic advection-diffusion-decay model

Abstract

Effects of non-stationarity on the performance of hybrid ensemble filters are studied (by hybrid filters we mean those which blend ensemble covariances with some other regularizing covariances). To isolate effects of non-stationarity from effects due to nonlinearity (and the non-Gaussianity it causes), a new doubly stochastic advection-diffusion-decay model (DSADM) is proposed. The model is hierarchical: it is a linear stochastic partial differential equation whose coefficients are random fields defined through their own stochastic partial differential equations. DSADM generates conditionally Gaussian spatiotemporal random fields with a tunable degree of non-stationarity in space and time. DSADM allows the use of the exact Kalman filter as a baseline benchmark. In numerical experiments with DSADM as the "model of truth", the relative importance of the three kinds of covariance blending is studied: with static, time-smoothed, and space-smoothed covariances. It is shown that the stronger the non-stationarity, the less useful the static covariance matrix becomes and the more beneficial the time-smoothed covariances are. Time-smoothing of background-error covariances proved to be systematically more useful than their space-smoothing. Under non-stationarity, a filter that extends the (previously proposed by the authors) Hierarchical Bayes Ensemble Filter and accommodates the three covariance-blending techniques is shown to outperform all other configurations of the filters tested. The R code of the model and the filters is available from github.com/cyrulnic/NoStRa.