Quantifying and visualizing uncertainty of 3D geological structures with implicit methods
Keywords: Geological Sciences.

Geological structures significantly contribute the complex interaction of physical processes in subsurface systems. The evaluation of the spatial distribution of geological structures in the subsurface are crucial for various applications, so sophisticated methods are needed to model and visualize geological structures in 3D. However, uncertainties are unavoidable for these 3D models, due to sparsity and imprecision of data, as well as people's lack of geological understanding. Both methodological and computational challenges exist in addressing uncertainties of 3D geological structures. This dissertation addresses these challenges, by presenting new practical methods for quantifying and visualizing the uncertainty of geological structures with implicit methods. To enhance people's communication and perception about structural uncertainty, a new method based on the idea of stochastic motion is proposed first. Geological surfaces are represented as the addition of trend functions, initialized with signed distance functions, and residual functions, subject to constraints of data and geological age relationships. The uncertainty is assessed by independent realizations drawn by Monte Carlo sampling. The uncertainty is visualized by a "smooth" movie of gradually evolving geological surfaces that have the same stationary distribution as Monte Carlo realizations, sampled by McMC. The method is illustrated using a synthetic data set from a copper deposit, where denser drillholes constrain an ore body with seven different lithologies. For handling more complex cases with even denser data and more geological rules, a framework to model large-scale geological structures is presented. Due to the non-stationary and complex nature of large-scale geological structures, performing global interpolation with all dense data together may create artifacts that are geologically unrealistic. Therefore, the proposed framework uses a divide-and-conquer strategy. The core idea is to create intermediate implicit 3D geological models that match subsets of data and then recombine them into a single large 3D geological model, while maintaining data and geological rule constraints. The framework is successfully applied to model the stratigraphy model of a large-scale banded iron formation in Western Australia with dense boreholes. Finally, an efficient Bayesian framework to quantify the uncertainty of implicit geological structures with geophysical data is introduced. Geophysical data provide critical information and constraints for validating subsurface models. Bayesian frameworks are often needed for quantifying uncertainty of 3D geological structures in inverse problems, but challenges exist, due to the high dimensional nature of spatial models. Implicit representation of geological structures transforms discrete geological objects into a continuous variable, i.e., a scalar field; dimension reduction techniques such as principal component analysis can be applied because of the implicit representation. Rejection sampling and Metropolis-Hastings sampling algorithms are designed to work in the case. Results show that computing time is saved when sampling new model realizations from the low dimensional space. The method is demonstrated with a mineral-hosting region in Western Australia with gravity data