Implications of Stochastic and Deterministic Filters as Ensemble-Based Data Assimilation Methods in Varying Regimes of Error Growth

Accurate numerical prediction of fluid flows requires accurate initial conditions. Monte Carlo methods have become a popular and realizable approach to estimating the initial conditions necessary for forecasting, and have generally been divided into two classes: stochastic filters and deterministic filters. Both filters strive to achieve the error statistics predicted by optimal linear estimation, but accomplish their goal in different fashions, the former by way of random number realizations and the latter via explicit mathematical transformations. Inspection of the update process of each filter in a one-dimensional example and in a two-dimensional dynamical system offers a geometric interpretation of how their behavior changes as nonlinearity becomes appreciable. This interpretation is linked to three ensemble assessment diagnostics: rms analysis error, ensemble rank histograms, and measures of ensemble skewness and kurtosis. Similar expressions of these diagnostics exist in a hierarchy of models. The geometric interpretation and the ensemble diagnostics suggest that both filters perform as expected in a linear regime, but that stochastic filters can better withstand regimes with nonlinear error growth.