  Ensemble smoother (ES) has been widely used in various research fields to
reduce the uncertainty of the system-of-interest. However, the commonly-adopted
ES method that employs the Kalman formula, that is, ES$_\text{(K)}$, does not
perform well when the probability distributions involved are non-Gaussian. To
address this issue, we suggest to use deep learning (DL) to derive an
alternative update scheme for ES in complex data assimilation applications.
Here we show that the DL-based ES method, that is, ES$_\text{(DL)}$, is more
general and flexible. In this new update scheme, a high volume of training data
are generated from a relatively small-sized ensemble of model parameters and
simulation outputs, and possible non-Gaussian features can be preserved in the
training data and captured by an adequate DL model. This new variant of ES is
tested in two subsurface characterization problems with or without Gaussian
assumptions. Results indicate that ES$_\text{(DL)}$ can produce similar (in the
Gaussian case) or even better (in the non-Gaussian case) results compared to
those from ES$_\text{(K)}$. The success of ES$_\text{(DL)}$ comes from the
power of DL in extracting complex (including non-Gaussian) features and
learning nonlinear relationships from massive amounts of training data.
Although in this work we only apply the ES$_\text{(DL)}$ method in parameter
estimation problems, the proposed idea can be conveniently extended to analysis
of model structural uncertainty and state estimation in real-time forecasting
studies.
