  The 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, 1000, 2000) and seven EnKF-variants (Damped,
Iterative, Local, Hybrid, Dual, Normal Score and Classical EnKF), we computed
1000 synthetic parameter estimation experiments for two set-ups: a 2D tracer
transport problem and a 2D flow problem with one injection well. For each
model, the only difference among synthetic experiments was the generated set of
random permeability fields. The 1000 synthetic experiments allow to calculate
the pdf of the RMSE of the characterization of the permeability field.
Comparing mean RMSEs for different EnKF-variants, ensemble sizes and
flow/transport set-ups 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 set-up
and the ensemble size.
