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.

由于相对较低的计算成本和算法的并行性，迭代集成平滑器已广泛用于校准各种物理系统的模拟器。但是，在特定的物理模型和随后的离散数学模型具有准确模拟现实的能力的主要假设下，已经为完美模型设计了迭代合奏平滑器。尽管通常会付出巨大的努力来确保数学模型的准确性，但是众所周知，物理模型只是现实的近似。这些近似值通常会引入某种类型的模型误差，这通常是未知的，并且在对模型进行校准时，可以通过调整模型参数以匹配历史观测值来抹去模型误差的影响。这导致估计参数存在偏差，结果可能导致质量可疑的预测。在本文中，我们制定了一个灵活的迭代集成平滑器，可用于校准无法忽略模型误差的不完美模型。我们的方法基于具有多个数据同化的集成平滑器（ES-MDA），因为它是使用最广泛的迭代集成平滑技术之一。在提出的算法中，残差（数据不匹配）分为两部分。一部分用于导出参数更新，另一部分用于表示模型误差。所提出的方法相当笼统，并且放宽了文献中通常引入的许多假设。