The ensemble Kalman filter (EnKF) is a Monte Carlo method for data assimilation and assessment of uncertainties during reservoir characterization and performance forecasting. The method is based on a low-rank approximation to the system covariance matrix calculated from an ensemble which may be orders of magnitude smaller than the number of state variables. In practical applications, the ensemble size has to be kept relatively small. This may lead to poor approximation of the cross-covariance matrix, and sampling errors can result in spurious correlations and incorrect changes in the state variables. Also, since the rank of the covariance matrix cannot be larger than the number of ensemble members, the number of degrees of freedom may be too low when a large number of measurements are assimilated, such as with 4D seismic data. In this work, we have investigated the shortcomings of a straightforward EnKF implementation for small ensemble size, relative to a large number of measurements. This is done by considering a single update of a simple linear model and comparing the EnKF update to the traditional Kalman filter (or Kriging) solution, which in this case is exact. The quality of the EnKF update is assessed by considering the mean and variance of the updated state variable, as well as various error norms and the eigen-spectrum of the covariance matrix. Even for this simple model, spurious long-range correlation, ensemble collapse, etc. are clearly seen as the number of measurements increases for a given ensemble size. For a traditional implementation of EnKF, the ensemble size have to be much larger than the number of measurements to obtain an accurate solution, and the solution gets worse when the measurement uncertainty is reduced.


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