Abstract
Iterative ensemble smoothers have been widely used for calibrating simulators of various physical systems due to the relatively low computational cost and the parallel nature of the algorithm. However, iterative ensemble smoothers have been designed for perfect models under the main assumption that the specified physical models and subsequent discretized mathematical models have the capability to model the reality accurately. While significant efforts are usually made to ensure the accuracy of the mathematical model, it is widely known that the physical models are only an approximation of reality. These approximations commonly introduce some type of model error which is generally unknown and when the models are calibrated, the effects of the model errors could be smeared by adjusting the model parameters to match historical observations. This results in a bias estimated parameters and as a consequence might result in predictions with questionable quality. In this paper, we formulate a flexible iterative ensemble smoother, which can be used to calibrate imperfect models where model errors cannot be neglected. We base our method on the ensemble smoother with multiple data assimilation (ES-MDA) as it is one of the most widely used iterative ensemble smoothing techniques. In the proposed algorithm, the residual (data mismatch) is split into two parts. One part is used to derive the parameter update and the second part is used to represent the model error. The proposed method is quite general and relaxes many of the assumptions commonly introduced in the literature. We observe that the proposed algorithm has the capability to reduce the effect of model bias by capturing the unknown model errors, thus improving the quality of the estimated parameters and prediction capacity of imperfect physical models.
Highlights
Bayesian inversion is a generic inference framework that is widely adopted for calibration of mathematical models while accounting for different types/sources of uncertainties
After iteration 2, the proposed split parameter approached 1, which show that all the data mismatch is treated as a model error ensemble and further reduction of uncertainty is not possible due to the limited capacity of both the linear and quadratic models in matching the data. These results demonstrate the ability of the proposed algorithm in capturing the unknown model error uncertainties during the calibration of imperfect models
This flexible algorithm builds on the ensemble smoother with multiple data assimilation, which has the assumption that models are perfect, i.e., accurate representation of the real systems
Summary
Heriot–Watt University, Edinburgh, UK 2 Total Geoscience Research Centre, Aberdeen, UK. Ensemble-based methods are designed with the assumption that utilized mathematical model provides a complete representation of real physical systems and that the model errors are small enough that it could be neglected during the calibration process This assumption might introduce bias in the estimated parameter distribution [1, 26] and as a consequence results in bad quality predictions using the calibrated models. An iterative ensemble smoother is formulated with an approximate method to quantify the model errors during the calibration process in order to reduce and in some cases eliminate any bias in the estimated model parameters. In. ES-MDA, non-linear inverse problem is solved iteratively with an inflated noise covariance matrix and the inflation factor α is normally set to the total number of data assimilations/iterations Na. For the proposed Flexible ES-MDA, the output of the imperfect model is related to the perfect model output using the following equation:. The imperfect reservoir model has two sources of modeling errors, simplified geological representation and up-scaling errors
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