Abstract Ground-water flow and the transport of contaminants in groundwater systems are strongly controlled by geologic heterogeneities. Because, the data available to model these subsurface heterogeneities are generally sparse, the observed groundwater flow, piezometric head at well locations, or the concentration profile of solutes provide valuable additional information about these complex subsurface systems. Regardless, there is likely to be considerable uncertainty associated with the predictions of the subsurface heterogeneities. This has motivated the development of several ensemble-based schemes for assimilating flow response data into predictions of subsurface heterogeneities. Most ensemble-based data assimilation methods including ensemble Kalman filter (EnKF) or indicator-based data assimilation (InDA), used for assimilation of dynamic data into geologic models, utilize statistics in the form of covariances calculated using the ensemble of models to perform model updates. These covariance based updates do not preserve the complex spatial characteristics of geologic structures that are better represented using multiple-point statistics. Additionally, in EnKF, the mismatch between the observed and the simulated values are assumed to be linearly related to the parameter updates, and the distribution of the state variable is assumed to be multi-Gaussian. The spatial distribution of the primary variables like facies can be non-Gaussian because of the spatial continuity exhibited by channel facies (sands) and non-channel facies (clay), that have distinctly different properties. Also, the relationship between the mismatch and the parameter updates can be strongly non-linear. In indicator-based data assimilation (InDA) method the indicator transforms of parameter and mismatch variables are used, which affords the treatment of these variables as bi-variate non-Gaussian. Furthermore, in the indicator transformed space, restrictive linear assumption can also be lifted as the indicator transform is invariant under non-linear transformations. However, the updates are still governed by the bivariate interactions between the state variables. A multiple-point extension of InDA is proposed in this paper which utilizes existing multiple-point simulation algorithms like single normal equation simulation (SNESIM) in combination with InDA, to preserve the spatial characteristics of the geologic model while at the same time honor the observed flow or transport response. The proposed method is applied to a synthetic groundwater system with complex channel distributions. The channel features of the final updated ensemble of models is shown to converge towards the reference model and the ensemble flow responses are also shown to match the reference flow response.

中文翻译：

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基于指标的数据同化与多点统计，用于更新具有非高斯参数分布的模型集合

摘要 地下水流和地下水系统中污染物的迁移受到地质异质性的强烈控制。因为可用于模拟这些地下异质性的数据通常很少，所以观察到的地下水流、井位压力水头或溶质的浓度分布提供了有关这些复杂地下系统的有价值的附加信息。无论如何，地下异质性的预测可能存在相当大的不确定性。这促使开发了几种基于集合的方案，用于将流动响应数据同化为地下非均质性的预测。大多数基于集合的数据同化方法，包括集合卡尔曼滤波器 (EnKF) 或基于指标的数据同化 (InDA)，用于将动态数据同化到地质模型中，利用使用模型集合计算的协方差形式的统计数据来执行模型更新。这些基于协方差的更新不保留使用多点统计更好地表示的地质结构的复杂空间特征。此外，在 EnKF 中，假设观察值和模拟值之间的失配与参数更新线性相关，并且状态变量的分布假设为多高斯分布。由于河道相（砂）和非河道相（粘土）表现出的空间连续性，像相这样的主要变量的空间分布可能是非高斯分布的，它们具有明显不同的特性。还，失配和参数更新之间的关系可能是强非线性的。在基于指标的数据同化 (InDA) 方法中，使用了参数和失配变量的指标变换，这使得将这些变量作为双变量非高斯处理。此外，在指标变换空间中，由于指标变换在非线性变换下是不变的，因此也可以取消限制性线性假设。然而，更新仍然由状态变量之间的双变量交互控制。本文提出了 InDA 的多点扩展，它利用现有的多点模拟算法，如单正规方程模拟 (SNESIM) 结合 InDA，保留地质模型的空间特征，同时尊重观察到的流动或运输响应。所提出的方法应用于具有复杂渠道分布的合成地下水系统。最终更新的模型集合的通道特征显示为向参考模型收敛，并且集合流响应也显示为与参考流响应匹配。