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