Dilated Spatial Generative Adversarial Networks for Ergodic Image Generation

Generative models have recently received renewed attention as a result of adversarial learning. Generative adversarial networks consist of samples generation model and a discrimination model able to distinguish between genuine and synthetic samples. In combination with convolutional (for the discriminator) and de-convolutional (for the generator) layers, they are particularly suitable for image generation, especially of natural scenes. However, the presence of fully connected layers adds global dependencies in the generated images. This may lead to high and global variations in the generated sample for small local variations in the input noise. In this work we propose to use architec-tures based on fully convolutional networks (including among others dilated layers), architectures specifically designed to generate globally ergodic images, that is images without global dependencies. Conducted experiments reveal that these architectures are well suited for generating natural textures such as geologic structures

Variational Bayesian inference with complex geostatistical priors using inverse autoregressive flows

We combine inverse autoregressive flows (IAF) and variational Bayesian inference (variational Bayes) in the context of geophysical inversion parameterized with deep generative models encoding complex priors. Variational Bayes approximates the unnormalized posterior distribution parametrically within a given family of distributions by solving an optimization problem. Although prone to bias if the chosen family of distributions is too limited, it provides a computationally-efficient approach that scales well to high-dimensional inverse problems. To enhance the expressiveness of the variational distribution, we explore its combination with IAFs that allow samples from a simple base distribution to be pushed forward through a series of invertible transformations onto an approximate posterior. The IAF is learned by maximizing the lower bound of the evidence (marginal likelihood), which is equivalent to minimizing the Kullback–Leibler divergence between the approximation and the target posterior distribution. In our examples, we use either a deep generative adversarial network (GAN) or a variational autoencoder (VAE) to parameterize complex geostatistical priors. Although previous attempts to perform Gauss–Newton inversion in combination with GANs of the same architecture were proven unsuccessful, the trained IAF provides a good reconstruction of channelized subsurface models for both GAN- and VAE-based inversions using synthetic crosshole ground-penetrating-radar data. For the considered examples, the computational cost of our approach is seven times lower than for Markov chain Monte Carlo (MCMC) inversion. Furthermore, the VAE-based approximations in the latent space are in good agreement. The VAE-based inversion requires only one sample to estimate gradients with respect to the IAF parameters at each iteration, while the GAN-based inversions need more samples and the corresponding posterior approximation is less accurate

Estimation of spatially-structured subsurface parameters using variational autoencoders and gradient-based optimization

Environmental models of the subsurface usually require the estimation of high-dimensional spatially-distributed parameters. However, the sparsity of subsurface data hinders such estimation which may in turn affect predictions using these models. In order to mitigate this issue, parameter estimation can be constrained to prior information on the expected subsurface patterns (e.g. geological facies). By using several examples of such patterns (e.g. those obtained from a training image), a variational autoencoder (VAE) learns a low-dimensional latent space that can be seen as a reparameterization of the original high-dimensional parameters and then estimation by means of gradient-based optimization can be performed in this latent space. Spatial parameters estimated in this way display the enforced patterns. VAEs usually include deep neural networks within their architecture and have shown good performance in reproducing high-dimensional structured subsurface models. However, they use a highly nonlinear function to map from latent space to the original high-dimensional parameter space which may give rise to local minima where gradient-based optimization gets trapped and therefore fails to reach the global minimum. Global optimization strategies may be used to solve this issue, however, a gradient-based inversion is preferred because of its lower computational demand. We propose using VAE together with gradient-based optimization in a linear traveltime tomography synthetic case with added noise and show that it often reaches low data error values and produces visually similar spatial parameters when compared with different test subsurface realizations. We also add regularization to the objective function to improve the number of times it reaches an adequate minimum. Finally, we perform a synthetic test with nonlinear traveltime tomography and show that the proposed strategy is able to recover visually similar spatial parameters with an error close to the added noise level

A new framework to reduce uncertainty in Wellhead Protection Area prediction using Bayesian Evidential Learning

Decisions related to groundwater management such as sustainable extraction of drinking water or protection against contamination can have great socioeconomic impacts. Ideally, a complete uncertainty analysis should be performed to assess risk and foresee all outcomes of any prediction. Uncertainties stem from the intricacy of the subsurface physical properties and the scarcity of data, measuring directly or indirectly the parameters of interest, relying on our limited understanding of the physical processes at stake. In this contribution, we compute the wellhead protection area (WHPA), whose shape and extent is influenced by the hydraulic conductivity (K) distribution, from tracing experiments. We predict WHPA within the Bayesian Evidential Learning framework, which aims at finding a direct relationship between data (d, tracing experiments) and forecast (f, WHPA) using machine learning. Given n different K fields describing n unconfined aquifer models with one pumping well, 2n forward models are computed to generate n samples of d and f. Both their dimensions are reduced using Principal Component Analysis, and a relationship between the two is sought using Canonical Correlation Analysis, allowing us to infer the mean and posterior covariance of the prediction in reduced space. Drawing m random samples from this multivariate normal distribution and back-transforming them in the original physical space generates m posterior WHPA to quantify the range of uncertainty in the prediction. This range is affected by the number and position of tracer wells. The aim is then to optimize the design of those wells to reduce the uncertainty. Increasing the number of injecting wells within the WHPA effectively does so, as the breakthrough curves will store information on a larger portion of the K field surrounding the pumping well. Experimental design prior to field investigation as proposed in our approach helps to identify the optimal location of wells, within limited budget constraints