The Data Assimilation Approach in a Multilayered Uncertainty Space

Martin Drieschner, Clemens Herrmann, Yuri Petryna
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Abstract

The simultaneous consideration of a numerical model and of different observations can be achieved using data-assimilation methods. In this contribution, the ensemble Kalman filter (EnKF) is applied to obtain the system-state development and also an estimation of unknown model parameters. An extension of the Kalman filter used is presented for the case of uncertain model parameters, which should not or cannot be estimated due to a lack of necessary measurements. It is shown that incorrectly assumed probability density functions for present uncertainties adversely affect the model parameter to be estimated. Therefore, the problem is embedded in a multilayered uncertainty space consisting of the stochastic space, the interval space, and the fuzzy space. Then, we propose classifying all present uncertainties into aleatory and epistemic ones. Aleatorically uncertain parameters can be used directly within the EnKF without an increase in computational costs and without the necessity of additional methods for the output evaluation. Epistemically uncertain parameters cannot be integrated into the classical EnKF procedure, so a multilayered uncertainty space is defined, leading to inevitable higher computational costs. Various possibilities for uncertainty quantification based on probability and possibility theory are shown, and the influence on the results is analyzed in an academic example. Here, uncertainties in the initial conditions are of less importance compared to uncertainties in system parameters that continuously influence the system state and the model parameter estimation. Finally, the proposed extension using a multilayered uncertainty space is applied on a multi-degree-of-freedom (MDOF) laboratory structure: a beam made of stainless steel with synthetic data or real measured data of vertical accelerations. Young’s modulus as a model parameter can be estimated in a reasonable range, independently of the measurement data generation.
多层不确定性空间中的数据同化方法
利用资料同化方法可以同时考虑一个数值模型和不同的观测值。在这篇贡献中,集成卡尔曼滤波器(EnKF)被用于获得系统状态发展和未知模型参数的估计。对于模型参数不确定的情况,由于缺乏必要的测量,不应该或不能估计,提出了卡尔曼滤波器的扩展。结果表明,不正确地假设当前不确定性的概率密度函数会对待估计的模型参数产生不利影响。因此,该问题被嵌入到由随机空间、区间空间和模糊空间组成的多层不确定性空间中。然后,我们建议将所有存在的不确定性分为选择性不确定性和认识性不确定性。任意不确定参数可以直接在EnKF中使用,而不会增加计算成本,也不需要额外的输出评估方法。传统的EnKF算法无法将认知上的不确定性参数整合到该算法中,因此需要定义一个多层的不确定性空间,这必然会带来较高的计算成本。给出了基于概率论和可能性理论的不确定性量化的各种可能性,并通过一个学术实例分析了不确定性量化对结果的影响。与持续影响系统状态和模型参数估计的系统参数的不确定性相比,初始条件中的不确定性不那么重要。最后,利用多层不确定性空间提出的扩展应用于多自由度(MDOF)实验室结构:由不锈钢制成的梁,具有垂直加速度的合成数据或实际测量数据。杨氏模量作为模型参数可以在一个合理的范围内估计,而不依赖于测量数据的产生。
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