Geophysical methods, such as electrical resistivity tomography (ERT), can be used to imaging the near-surface electrical resistivity as field measurements depend on the subsurface porosity, water saturation and fluid salinity. ERT has been widely applied to investigate mineral and groundwater resources, and in archaeological, environmental, and engineering studies. The prediction of subsurface electrical conductivity from ERT data requires solving a geophysical inverse problem. For near-surface characterization studies, this is often accomplished with deterministic inverse methods. These methods linearize the problem around an initial solution, and their smoothness depends on an imposed a priori spatial regularization term. Depending on this parameterization, these methodologies might struggle to capture the natural variability of the subsurface. Moreover, deterministic solutions have limited capabilities for uncertainty assessment. On the contrary, stochastic inverse methods can assess uncertainties by predicting multiple model realizations that fit similarly the recorded ERT data. However, they are often more computationally expensive than deterministic solutions. Deep learning algorithms based on deep generative models have been used to re-parametrize model and data spaces into low-dimensional domains and efficiently solve geophysical inverse problems. However, within this context, uncertainty assessment is challenging. We propose a deep convolutional variational autoencoder (VAE) coupled with stochastic adaptive optimization to perform stochastic ERT inversion. Geostatistical simulations of electrical resistivity are used as training data set of the VAE. After training, the VAE generates electrical resistivity models that reproduce the statistics and spatial continuity patterns of the training data set. Then, the VAE latent space is iteratively perturbed and updated with adaptive stochastic sampling based on the misfit between observed and predicted ERT data. The proposed methodology is illustrated in two-dimensional synthetic and real data sets to illustrate the ability of the proposed method to predict reliable electrical resistivity models while generating multiple possible scenarios for uncertainty assessment.
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