Estimating spatially distributed properties such as hydraulic conductivity (K) from available sparse measurements is a great challenge in subsurface characterization. However, the use of inverse modeling is limited for ill-posed, high-dimensional applications due to computational costs and poor prediction accuracy with sparse datasets. In this paper, we combine Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP), a deep generative model that can accurately capture complex subsurface structure, and Ensemble Smoother with Multiple Data Assimilation (ES-MDA), an ensemble-based inversion method, for accurate and accelerated subsurface characterization. WGAN-GP is trained to generate high-dimensional K fields from a low-dimensional latent space and ES-MDA then updates the latent variables by assimilating available measurements. Several subsurface examples are used to evaluate the accuracy and efficiency of the proposed method and the main features of the unknown K fields are characterized accurately with reliable uncertainty quantification. Furthermore, the estimation performance is compared with a widely-used variational, i.e., optimization-based, inversion approach, and the proposed approach outperforms the variational inversion method, especially for the channelized and fractured field examples. We explain such superior performance by visualizing the objective function in the latent space: because of nonlinear and aggressive dimension reduction via generative modeling, the objective function surface becomes extremely complex while the ensemble approximation can smooth out the multi-modal surface during the minimization. This suggests that the ensemble-based approach works well over the variational approach when combined with deep generative models at the cost of forward model runs unless convergence-ensuring modifications are implemented in the variational inversion.

翻译：从有限的稀疏测量数据估计水力传导度（K）等空间分布属性是subsurface表征中的重大挑战。然而，由于计算成本高且稀疏数据集预测精度差，反演建模在病态高维问题中的应用受到限制。本文结合Wasserstein生成对抗网络与梯度惩罚（WGAN-GP，一种能精确捕捉复杂subsurface结构的深度生成模型）和集成平滑器与多次数据同化（ES-MDA，一种基于集成的反演方法），以实现精确且加速的subsurface表征。训练WGAN-GP从低维潜空间生成高维K场，然后利用ES-MDA通过同化可用观测数据更新潜变量。通过多个subsurface实例评估所提方法的准确性和效率，未知K场的主要特征均被准确表征并附有可靠的不确定性量化。此外，与广泛使用的变分（即基于优化的）反演方法进行性能对比，所提方法在裂缝性和裂隙场实例中尤其优于变分反演方法。通过可视化潜空间中的目标函数解释这种优越性能：由于生成模型通过非线性且激进的降维，目标函数曲面变得极其复杂，而集成近似能在最小化过程中平滑多峰曲面。这表明，当与深度生成模型结合时，除非在变分反演中实施收敛保证性修正，否则基于集成的方法在以正演模型运行次数为代价的情况下，其性能优于变分方法。