In geological characterization, the traditional methods that rely on the covariance matrix for continuous variable estimation often either neglect or oversimplify the challenge posed by subsurface non-stationarity. This study presents an innovative methodology using ancillary data such as geological insights and geophysical exploration to address this challenge directly, with the goal of accurately delineating the spatial distribution of subsurface petrophysical properties, especially, in large geological fields where non-stationarity is prevalent. This methodology is based on the geodesic distance on an embedded manifold and is complemented by the level-set curve as a key tool for relating the observed geological structures to intrinsic geological non-stationarity. During validation, parameters 𝜌 and 𝛽 were revealed to be the critical parameters that influenced the strength and dependence of the estimated spatial variables on secondary data, respectively. Comparative evaluations showed that our approach performed better than a traditional method (i.e., kriging), particularly, in accurately representing the complex and realistic subsurface structures. The proposed method offers improved accuracy, which is essential for high-stakes applications such as contaminant remediation and underground repository design. This study focused primarily on twodimensional models. There is a need for three-dimensional advancements and evaluations across diverse geological structures. Overall, this research presents novel strategies for estimating non-stationary geologic media, setting the stage for improved exploration of subsurface characterization in the future.

In this study, the combined application of geodesic kernel and Gaussian process regression was investigated to estimate nonstationary hydraulic conductivity fields in two-dimensional hydrogeological systems. Particularly, a semi-analytical form of the geodesic distance based on the intrinsic geometry of the manifold was derived and used to define positive definite geodesic covariance matrices that are employed for Gaussian process regression. Furthermore, the proposed approach was applied to a series of synthetic hydraulic conductivity estimation problems and the results show that the incorporation of secondary information, such as geophysical or geological interpretations, can considerably improve the estimation accuracy, especially in nonstationary fields. Moreover, groundwater flow and solute transport simulations based on the estimated hydraulic conductivity fields revealed that the accuracy of the simulations was strongly affected by the inclusion of secondary information. These results suggest that incorporating secondary information into manifold geometry can remarkably improve the estimation accuracy and provide new insights on the underlying structure of geological data. This proposed approach has crucial implications for hydrogeological applications, such as groundwater resource management, safety assessments, and risk management strategies related to groundwater contamination.

A document by Eungyu Park. Click on the document to view its contents.