Comparing Seven Variants of the Ensemble Kalman Filter: How Many Synthetic Experiments Are Needed?

Abstract

AbstractThe ensemble Kalman filter (EnKF) is a popular estimation technique in the geosciences. It is used as a numerical tool for state vector prognosis and parameter estimation. The EnKF can, for example, help to evaluate the geothermal potential of an aquifer. In such applications, the EnKF is often used with small or medium ensemble sizes. It is therefore of interest to characterize the EnKF behavior for these ensemble sizes. For seven ensemble sizes (50, 70, 100, 250, 500, 1,000, and 2,000) and seven EnKF variants (damped, iterative, local, hybrid, dual, normal score, and classical EnKF), we computed 1,000 synthetic parameter estimation experiments for two setups: a 2‐D tracer transport problem and a 2‐D flow problem with one injection well. For each model, the only difference among synthetic experiments was the generated set of random permeability fields. The 1,000 synthetic experiments allow to calculate the probability density function of the root‐mean‐square error (RMSE) of the characterization of the permeability field. Comparing mean RMSEs for different EnKF variants, ensemble sizes and flow/transport setups suggests that multiple synthetic experiments are needed for a solid performance comparison. In this work, 10 synthetic experiments were needed to correctly distinguish RMSE differences between EnKF variants smaller than 10%. For detecting RMSE differences smaller than 2%, 100 synthetic experiments were needed for ensemble sizes 50, 70, 100, and 250. The overall ranking of the EnKF variants is strongly dependent on the physical model setup and the ensemble size.