Manifold embedding of geological and geophysical observations for non-stationary subsurface property estimation using geodesic Gaussian processes

Traditional methods for geological characterization often overlook or oversimplify the challenge of subsurface non-stationarity. This study introduces an innovative methodology that uses ancillary data, such as geological insights and geophysical exploration, to accurately delineate the spatial distribution of subsurface petrophysical properties in large, non-stationary geological fields. The approach leverages geodesic distance on an embedded manifold, with the level-set curve linking observed geological structures to intrinsic non-stationarity. Critical parameters $\rho$ and $\beta$ were identified, influencing the strength and dependence of estimates on secondary data. Comparative evaluations demonstrated that this method outperforms traditional kriging, particularly in representing complex subsurface structures. This enhanced accuracy is crucial for applications such as contaminant remediation and underground repository design. While focused on two-dimensional models, future work should explore three-dimensional applications across diverse geological structures. This research provides novel strategies for estimating non-stationary geologic media, advancing subsurface characterization.