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This thesis examines the use of generative adversarial networks (GANs) as a parameterization tool for inverse problems solved with ensemble-based data assimilation methods. Ensemble methods often rely on the assumption of Gaussian distributed parameters in cases where this assumption is not valid, the parameter estimation can be invalid. Parameterization methods allow the transformation of these non-Gaussian parameters into a better suited distribution, and optimally reduce their dimension.
Another limitation of ensemble methods is the injection of prior information of the physical relation as a constraint between parameters such as spatial coherence or physical balances. Optimal parameterization should encompass these different properties to facilitate the estimation. The novel approach presented in this work relies on GANs to achieve these objectives. Two application domains are tackled through the present work.

In a first application, subsurface reservoir characterization, the objective is to determine geological properties of a numerical reservoir model from the observation of the reservoir dynamical response by the way of data assimilation. Rock facies, that describe the type of rock present in each cell of the numerical model, have to be determined due to their strong influence on the dynamical response. The rock facies spatial distribution is ruled by geological phenomena such as sedimentation and forms well known patterns, like channels, called heterogeneities. The non-continuous property and their spatial coherence make their characterization by ensemble-based data assimilation algorithms difficult, and requires parameterization. Parameterization is a challenge for numerous heterogeneities, notably channels, due to the numerical cost or the statistical representation of their spatial distribution.

A Second application domain is the atmospheric balance in the context of numerical weather prediction. When new observations are available, correction of the atmospheric state is done using ensemble based data assimilation methods. This correction step can introduce imbalance in the physical state and cause numerical instability during the integration in time of the atmosphere, deteriorating the information brought by the previous observations. The importance of generating or correcting balanced climate, also called initialized atmospheric state, during data assimilation is then a key step in numerical weather prediction.

This work aims at presenting the performance of GAN parameterization and its multi-disciplinary applicability to researchers who are not familiar with the domain of deep learning. GAN is an unsupervised deep learning method belonging to the deep generative network family, able to learn a dataset distribution and generate new samples from the learned distribution in an unsupervised way. These synthetic samples are encoded in a low-dimensional latent space that can be sampled from a Gaussian distribution that is suited to perform ensemble data assimilation. Their recent ability to generate complex images led us to consider them as a good candidate for parameterization method. The unsupervised property of this type of parameterization makes it applicable to several diverse domains such as learning the pattern of geological heterogeneities or learning the physical constraints that makes an atmospheric state balanced.

This study shows how to train GANs for two different applications : subsurface reservoir and climate data. The use of the parameterization in an ensemble based data assimilation such as ensemble smoother with multiple data assimilation (ES-MDA) is demonstrated for subsurface reservoirs. Finally, a posteriori conditioning of the GAN function is examined using derivative free optimization.

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