Quantifying microstructures of Earth materials: Reconstructing higher-order correlation functions using deep generative adversarial networks

Key to most subsurface processes is to determine how structural and topological features at small length scales, i.e., the microstructure, control the effective and macroscopic properties of earth materials. Recent progress in imaging technology has enabled us to visualise and characterise microstructures at different length scales and dimensions. An approach to characterisation is the sampling of multi-point spatial correlation functions - known as statistical microstructural descriptors (SMDs) - from images. SMDs can then be used to generate statistically equivalent structures having different scales and additional dimensions – this process is known as $reconstruction$. We use higher-order SMDs ($n$-point polytope functions) to characterise two hydrothermally altered rocks as examples of quantifying varying degrees of geometric complexity in the Earth's lithosphere. Using a generative adversarial network (GAN), trained with Wasserstein-loss and gradient penalty, we subsequently reconstruct two-dimensional electron microscopy images of these rocks. In addition to improving the training stability of GANs, we show that our model is capable of reconstructing higher-order, spatially-correlated patterns of complex earth materials, capturing underlying structural and morphological properties. Our approach is critical to address a wide range of geoscientific challenges aiming at reconstructing morphology-dependent physical rock properties to, e.g.,: i) increase the number of digital samples from limited real samples to, e.g., assess the variability of rock transport properties; ii) perform upscaling, i.e., generate representative domains from high-resolution images of small complex samples and couple microstructures to macroscopic phenomena; iii) reconstruct 3D microstructures when only 2D images are available.