Time-lapse seismic or electromagnetic data are similar to production data in that they are generally strongly sensitive to averages of porosity and permeability over extended regions, but only weakly sensitive to the properties of individual gridblocks. Consequently, updates of model parameters from the ensemble Kalman filter (EnKF) will be subject to substantial error due to spurious correlations when the size of the ensemble is small. Adaptive localization methods will reduce the magnitude of the spurious correlations, but may also eliminate the true, but weak, correlations. For this type of data, there appears to be a relationship between the spatial scale of the model variables and the sensitivity of observations to the variables. By introducing a multiscale expansion with adaptive localization, we are able to increase the sensitivity to the larger scale variables and more accurately compute their magnitudes, while eliminating updates of the small scale variables with weak correlations.

In this paper we compare three approaches to data assimilation for observation operators with weak, nonlocal sensitivity to model variables. The first approach is the EnKF without localization of any kind. The second approach is the EnKF with adaptive bootstrap localization or screening of the Kalman gain. In the third approach, we apply the EnKF with adaptive bootstrap localization to the problem of updating a multiscale expansion of the model variables. These approaches are illustrated through application to several spatially distributed data assimilation problems. The first application is a one-dimensional vector of correlated random variables for which the observations are averages over intervals. The second example is a two-dimensional correlated array of permeability values for which the data are observations of pressure in an observation well at two times. The final example is also a two-dimensional flow problem, but one in which the concentration of an injectant is observed at several times. In the first two examples, the ensemble Kalman filter approach with a multiscale expansion and adaptive localization gave the best results. In the final example, results from adaptive localization with and without the multiscale expansion are essentially equivalent because the correlation of data limits the scale of the features in the Kalman gain matrix.

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