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In ensemble-based data assimilation (DA), the ensemble size is usually limited to around one hundred. Straightforward application of ensemble-based DA can therefore result in significant Monte Carlo errors, often manifesting themselves as severe underestimation of parameter uncertainties. Localization is the conventional remedy for this problem. Assimilation of large amounts of simultaneous data enhances the negative effects of Monte Carlo errors. Use of lower-fidelity models reduces the computational cost per ensemble member and therefore renders the possibility to reduce Monte Carlo errors by increasing the ensemble size, but it also adds to the modeling error. Multilevel data assimilation (MLDA) uses a selection of models forming hierarchies of both computational cost and computational accuracy, and tries to balance between Monte Carlo errors and modeling errors. In this work, we assess a recently developed MLDA algorithm, the Multilevel Hybrid Ensemble Smoother (MLHES), and introduce and assess an iterative version of this algorithm, the Iterative Multilevel Hybrid Ensemble Smoother (IMLHES). In our assessments, we compare these algorithms with conventional single-level DA algorithms with localization. To this end, a typical example of large amount of spatially distributed data, i.e. inverted seismic data, is considered and three data sets of this kind are assimilated in three different petroleum reservoir models. Qualitatively evaluating the DA outcomes, it is found that multilevel algorithms outperform their conventional single-level counterparts in obtaining the posterior statistics of both uncertain parameters and model forecasts. Additionally, it is observed that IMLHES performs better than MLHES in the same regard, and also successfully converges to the proximity of solution in a case where the considered iterative single-level algorithm did not converge to the global optimum.

With large amounts of simultaneous data, like inverted seismic data in reservoir modeling, negative effects of Monte Carlo errors in straightforward ensemble-based data assimilation (DA) are enhanced, typically resulting in underestimation of parameter uncertainties. Utilization of lower fidelity reservoir simulations reduces the computational cost per ensemble member, thereby rendering the possibility of increasing the ensemble size without increasing the total computational cost. Increasing the ensemble size will reduce Monte Carlo errors and therefore benefit DA results. The use of lower fidelity reservoir models will however introduce modeling errors in addition to those already present in conventional fidelity simulation results. Multilevel simulations utilize a selection of models for the same entity that constitute hierarchies both in fidelities and computational costs. In this work, we estimate and approximately account for the multilevel modeling error (MLME), that is, the part of the total modeling error that is caused by using a multilevel model hierarchy, instead of a single conventional model to calculate model forecasts. To this end, four computationally inexpensive approximate MLME correction schemes are considered, and their abilities to correct the multilevel model forecasts for reservoir models with different types of MLME are assessed. The numerical results show a consistent ranking of the MLME correction schemes. Additionally, we assess the performances of the different MLME-corrected model forecasts in assimilation of inverted seismic data. The posterior parameter estimates from multilevel DA with and without MLME correction are compared to results obtained from conventional single-level DA with localization. It is found that multilevel DA (MLDA) with and without MLME correction outperforms conventional DA with localization. The use of all four MLME correction schemes results in posterior parameter estimates with similar quality. Results obtained with MLDA without any MLME correction were also of similar quality, indicating some robustness of MLDA toward MLME.