Probabilistic Characterization of 3D Non-Stationary Spatial Variability from Limited Boreholes Using Bayesian Supervised Learning

Natural geomaterials are affected by many spatially varying factors during complex geological formation process and exhibit spatial variability and heterogeneity in three-dimensional (3D) space. Real geotechnical structures always interact with geomaterials with 3D spatial variability. However, in engineering practice, due to the limitation of investigation borehole number and the complex nature of geomaterials properties, it is challenging to fully characterize the 3D spatial variability. This challenge can be exacerbated when the layout of sparse boreholes has irregular spacing and the concerned spatial variability shows highly non-linear non-stationary features. Even the commonly used conventional geostatistical tools can hardly tackle this situation because the associated parameters, e.g., 3D non-stationary trend function, correlation structures along three different directions, are difficult to determine in the presence of limited borehole data. This paper aims to present a novel Bayesian supervised learning (BSL) method to resolve this challenge. Characterization of 3D spatial variability is formulated as a supervised learning problem and is solved under Bayesian framework. Interpretation uncertainty is quantified automatically. The BSL is non-parametric and data-driven. It bypasses the trend function determination and correlation structures estimation processes. The performance of BSL can evolve with the borehole number. The BSL method is illustrated using a highly non-linear non-stationary and anisotropic example. The results show the BSL can properly learn the spatial variability from limited borehole data and quantify associated uncertainty. By leveraging the sparsity-enhanced framework of BSL, the computational effort in 3D is feasible for personal computer.

Keywords: spatial variability, digital geological models, sparse data, machine learning, compressed sensing.