Fast stochastic inversion of geological interfaces from geophysical data using Bayesian Evidential Learning with Mixture Density Network

The subsurface is complex and heterogeneous, making its investigation very challenging. In this context, geophysical imaging methods can provide new insights on the spatial distribution of Earth physical properties. However, the imaging capacity of geophysical methods is limited by their indirect nature and the non-unicity of the solution of the inverse problem. In other words, the interpretation of geophysical data sets remains limited by their intrinsic uncertainty. On the one hand, deterministic methods fail to properly account for uncertainty. On the other hands, probabilistic approaches allowing uncertainty quantification, such as Markov chain Monte Carlo (McMC) methods, are both time-consuming and difficult to tune to convergence in complex subsurface systems with many parameters. Some alternatives, such as Bayesian Evidential Learning (BEL), providing an approximation of the posterior distribution have been proposed. BEL learns a statistical relationship between data and model parameters from a training set sampled from the prior distribution. This prevents the use of forward models during the prediction of the posterior distribution. The applications of probabilistic approaches to geophysical imaging often supposes simplifications in the distribution of model parameters to reduce the number of parameters.

In this contribution, we acknowledge that geophysical imaging is often not the objective of geophysical data acquisition. Geologists are often more interested in some specific features such as the depth of the bedrock, the location and geometry of a fault, or the spatial variability of the fresh-saltwater interface. We therefore define the prior distribution of model parameters in a hierarchical way, where the feature of interest is defined first with hyperparameters, explicitly included during posterior inference. This approach allows to decouple the imaging process from any pre-defined inversion grid. We use BEL to calculate the posterior distribution. To deal with the strong non-linearity of the data-model relationship, we use a mixture-density network with two hidden layers allowing to estimate the posterior distribution of model parameters.

We demonstrate the approach on a synthetic electrical resistivity tomography (ERT) example in a saline context. The fresh-saltwater interface is characterized using a third degree polynomial (4 parameters) separating a saltwater aquifer from an overlying freshwater lens, both with uncertain electrical resistivity. 10000 models are sampled from the prior distribution to train the BEL-MDN model between the ERT pseudo-section and the model parameters. PyGimLi is used to solve the ERT forward problem. During MDN training, the first epochs use noise free data; noisy data are only introduced later in the training process, allowing to maximize learning efficiency. Comparison with McMC shows that BEL-MDN is successful in identifying the depth and shape of the interface at a fraction of the cost of McMC. However, BEL-MDN tends to overestimate the uncertainty when the interface lies at shallow depth, which requires further research. The method holds great potential to image specific (hydro-)geological features, especially for complex cases where McMC are too computationally expensive.